Local unitary transformation, long-range quantum
entanglement, wave function renormalization, and
topological order
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Chen, Xie, Zheng-Cheng Gu, and Xiao-Gang Wen. "Local unitary
transformation, long-range quantum entanglement, wave
function renormalization, and topological order." Phys. Rev. B 82,
155138 (2010) [28 pages] © 2010 The American Physical
Society.
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http://dx.doi.org/10.1103/PhysRevB.82.155138
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Detailed Terms
PHYSICAL REVIEW B 82, 155138 共2010兲
Local unitary transformation, long-range quantum entanglement, wave function renormalization,
and topological order
Xie Chen,1 Zheng-Cheng Gu,2 and Xiao-Gang Wen1
1
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
共Received 28 July 2010; revised manuscript received 21 September 2010; published 26 October 2010兲
Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives
rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states
remain within the same phase under local unitary transformations. Therefore, local unitary transformations
define an equivalence relation and the equivalence classes are the universality classes that define the different
phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the
above equivalence/universality classes correspond to pattern of long-range entanglement, which is the essence
of topological order. The local unitary transformation also allows us to define a wave function renormalization
scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class.
Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary
transformations have finite dimensions. The solutions of the conditions allow us to classify this type of
topological orders, which generalize the string-net classification of topological orders. We also describe an
algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us
to calculate the flow of tensor-product wave functions which are not at the fixed points. This will allow us to
calculate topological orders as well as symmetry-breaking orders in a generic tensor-product state.
DOI: 10.1103/PhysRevB.82.155138
PACS number共s兲: 64.70.Tg, 71.27.⫹a
I. INTRODUCTION
According to the principle of emergence, the rich properties and the many different forms of materials originate from
the different ways in which the atoms are ordered in the
materials. Landau symmetry-breaking 共SB兲 theory provides a
general understanding of those different orders and resulting
rich states of matter.1,2 It points out that different orders really correspond to different symmetries in the organizations
of the constituent atoms. As a material changes from one
order to another order 共i.e., as the material undergoes a phase
transition兲, what happens is that the symmetry of the organization of the atoms changes.
For a long time, we believed that Landau symmetrybreaking theory describes all possible orders in materials and
all possible 共continuous兲 phase transitions. However, in last
20 years, it has become more and more clear that Landau
symmetry-breaking theory does not describe all possible orders. After the discovery of high-Tc superconductors in
1986,3 some theorists believed that quantum spin liquids play
a key role in understanding high-Tc superconductors4 and
started to introduce various spin liquids.5–9 Despite the success of Landau symmetry-breaking theory in describing all
kinds of states, the theory cannot explain and does not even
allow the existence of spin liquids. This leads many theorists
to doubt the very existence of spin liquids. In 1987, in an
attempt to explain high-temperature superconductivity, an infrared stable spin liquid—chiral spin state was
discovered,10,11 which was shown to be perturbatively stable
and exist as quantum phase of matter 共at least in a large N
limit兲. At first, not believing Landau symmetry-breaking
theory fails to describe spin liquids, people still wanted to
use symmetry breaking to describe the chiral spin state. They
identified the chiral spin state as a state that breaks the time1098-0121/2010/82共15兲/155138共28兲
reversal and parity symmetries but not the spin rotation
symmetry.11 However, it was quickly realized that there are
many different chiral spin states that have exactly the same
symmetry, so symmetry alone is not enough to characterize
different chiral spin states. This means that the chiral spin
states contain a new kind of order that is beyond symmetry
description.12 This new kind of order was named13 topological order.14
The key to identify 共and define兲 new orders is to identify
new universal properties that are beyond the local order parameters and long-range correlations used in the Landau
symmetry-breaking theory. Indeed, new quantum numbers,
such as ground-state degeneracy,12 the non-Abelian Berry’s
phase of degenerate ground states,13 and edge excitations,15
were introduced to characterize 共and define兲 the different topological orders in chiral spin states. Recently, it was shown
that topological orders can also be characterized by topological entanglement entropy.16,17 More importantly, those quantities were shown to be universal 共i.e., robust against any
local perturbation of the Hamiltonian兲 for chiral spin states.13
The existence of those universal properties establishes the
existence of topological order in chiral spin states.
Near the end of 1980s, the existence of chiral spin states
as a theoretical possibility, as well as their many amazing
properties, such as fractional statistics,10,11 spin-charge
separation,10,11 and chiral gapless edge excitations,15 were
established reliably, at least in the large N limit introduced in
Ref. 18. Even non-Abelian chiral spin states can be established reliably in the large N limit.19 However, it took about
10 years to establish the existence of a chiral spin state reliably without using large N limit 共based on an exactly soluble
model on honeycomb lattice兲.20
Soon after the introduction of chiral spin states, experiments indicated that high-temperature superconductors do
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PHYSICAL REVIEW B 82, 155138 共2010兲
CHEN, GU, AND WEN
not break the time-reversal and parity symmetries. So chiral
spin states do not describe high-temperature superconductors. Thus the theory of topological order became a theory
with no experimental realization. However, the similarity between chiral spin states and fractional quantum Hall 共FQH兲
states allows one to use the theory of topological order to
describe different FQH states.21 Just like chiral spin states,
different FQH states all have the same symmetry and are
beyond the Landau symmetry-breaking description. Also like
chiral spin states, FQH states have ground-state
degeneracies22 that depend on the topology of the space.13,21
Those ground-state degeneracies are shown to be robust
against any perturbations. Thus, the different orders in different quantum Hall states can be described by topological
orders, and the topological order does have experimental realizations.
The topology-dependent ground-state degeneracy, that
signal the presence of topological order, is an amazing phenomenon. In FQH states, the correlation of any local operators are short ranged. This seems to imply that FQH states
are “short sighted” and they cannot know the topology of
space which is a global and long-distance property. However,
the fact that ground-state degeneracy does depend on the
topology of space implies that FQH states are not short
sighted and they do find a way to know the global and longdistance structure of space. So, despite the short-range correlations of any local operators, the FQH states must contain
certain hidden long-range correlation. But what is this hidden
long-range correlation? This will be one of the main topic of
this paper.
Since high-Tc superconductors do not break the timereversal and parity symmetries, nor any other lattice symmetries, some people concentrated on finding spin liquids that
respect all those symmetries and hoping one of those symmetric spin liquids hold the key to understand high-Tc superconductors. Between 1987 and 1992, many symmetric spin
liquids were introduced and studied.5–9,23–26 The excitations
in some of constructed spin liquids have a finite energy gap
while in others there is no energy gap. Those symmetric spin
liquids do not break any symmetry and, by definition, are
beyond Landau symmetry-breaking description.
By construction, topological order only describes the organization of particles or spins in a gapped quantum state. So
the theory of topological order only applies to gapped spin
liquids. Indeed, we find that the gapped spin liquids do contain nontrivial topological orders25 共as signified by their
topology-dependent and robust ground-state degeneracies兲
and are described by topological quantum field theory 共such
as Z2 gauge theory兲 at low energies. One of the simplest
topologically ordered spin liquids is the Z2 spin liquid which
was first introduced in 1991.24,25 The existence of Z2 spin
liquid as a theoretical possibility, as well as its many amazing properties, such as spin-charge separation,24,25 fractional
mutual statistics,25 and topologically protected ground-state
degeneracy,25 were established reliably 共at least in the large
N limit introduced in Ref. 18兲. Later, Kitaev introduced the
famous toric code model which establishes the existence of
the Z2 spin liquid reliably without using large N limit.27 The
topologically protected degeneracy of the Z2 spin liquid was
used to perform fault-tolerant quantum computation.
The study of high-Tc superconductors also leads to many
gapless spin liquids. The stability and the existence of those
gapless spin liquid were in more doubts than their gapped
counterparts. But careful analysis in certain large N limit do
suggest that stable gapless spin liquids can exist.18,28,29 If we
do believe in the existence of gapless spin liquids, then the
next question is how to describe the orders 共i.e., the organizations of spins兲 in those gapless spin liquids. If gapped
quantum state can contain new type of orders that are beyond
Landau’s symmetry-breaking description, it is natural to expect that gapless quantum states can also contain new type of
orders. But how to show the existence of new orders in gapless states?
Just like topological order, the key to identify new orders
is to identify new universal properties that are beyond Landau symmetry description. Clearly we can no longer use the
ground-state degeneracy to establish the existence of new
orders in gapless states. To show the existence of new orders
in gapless states, a new universal quantity—projective symmetry group 共PSG兲—was introduced.30 It was argued that
共some兲 PSGs are robust against any local perturbations of the
Hamiltonian that do not change the symmetry of the
Hamiltonian.28–31 So through PSG, we establish the existence of new orders even in gapless states. The new orders
are called quantum order to indicate that the new orders are
related to patterns of quantum entanglement in the manybody ground state.32
II. SHORT-RANGE AND LONG-RANGE QUANTUM
ENTANGLEMENTS
What is missed in Landau theory so that it fails to describe those new orders? What is the new feature in the organization of particles/spins so that the resulting order cannot be described by symmetry?
To answer those questions, let us consider a simple
quantum system which can be described with Landau
theory—the transverse field Ising model in two dimensions:
H = −B兺Xi − J兺ZiZj, where Xi, Y i, and Zi are the Pauli matrices on site i. In B Ⰷ J limit, the ground state of the system is
an equal-weight superposition of all possible spin-up and
spin-down states: 兩⌽+典 = 兺兵␴i其兩兵␴i其典, where 兵␴i其 label a particular spin-up 共Zi = 1兲 and spin-down 共Zi = −1兲 configurations. In the J Ⰷ B limit, the system has two degenerate
ground states 兩⌽↑典 = 兩↑ ↑ ¯ ↑典 and 兩⌽↓典 = 兩↓ ↓ ¯ ↓典.
The transverse field Ising model has a Z → −Z symmetry.
The ground state 兩⌽+典 respect such a symmetry while the
ground state 兩⌽↑典 共or 兩⌽↓典兲 break the symmetry. Thus the
small J ground state 兩⌽+典 and the small B ground state 兩⌽↑典
describe different phases since they have different symmetries.
We note that 兩⌽↑典 is the exact ground state of the transverse field Ising model with B = 0. The state has no quantum
entanglement since 兩⌽↑典 is a direct product of local states:
兩⌽↑典 = 丢 i兩 ↑ 典i where 兩 ↑ 典i is an up-spin state at site i. The state
兩⌽+典 is the exact ground state of the transverse field Ising
model with J = 0. It is also a state with no quantum entanglement: 兩⌽+典 = 丢 i共兩 ↑ 典i + 兩 ↓ 典i兲 ⬀ 丢 i兩+典i, where 兩+典i ⬅ 兩 ↑ 典i + 兩 ↓ 典i is
a state with spin in x direction at site i.
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LOCAL UNITARY TRANSFORMATION, LONG-RANGE…
We see that the states 共or phases兲 described by Landau
symmetry-breaking theory has no quantum entanglement at
least in the J = 0 or B = 0 limits. In the J / B Ⰶ 1 and B / J Ⰶ 1
limits, the two ground states of the two limits still represent
two phases with different symmetries. However, in this case,
the ground states are not unentangled. On the other hand
since the ground states in the J = 0 or B = 0 limits have finite
energy gaps and short-range correlations, a small J or a small
B can only modify the states locally. Thus we expect the
ground states have only short-range entanglement.
The above example, if generalized to other symmetrybreaking states, suggests the following conjecture: if a
gapped quantum ground state is described by Landau
symmetry-breaking theory, then it has short-range quantum
entanglement.33
The direct-product states and short-range-entangled
共SRE兲 states only represent a small subset of all possible
quantum many-body states. Thus, according to the point of
view of the above conjecture, we see that what is missed by
Landau symmetry-breaking theory is long-range quantum
entanglement. It is this long-range quantum entanglement
that makes a state to have nontrivial topological/quantum
order.
However, mathematically speaking, the above conjecture
is a null statement since the meaning of short-range quantum
entanglement is not defined. In the following, we will try to
find a more precise description 共or definition兲 of short-range
and long-range quantum entanglements. We will start with a
careful discussion of quantum phases and quantum phase
transitions.
III. QUANTUM PHASES AND LOCAL UNITARY
EVOLUTIONS
To give a precise definition of quantum phases, let us
consider a local quantum system whose Hamiltonian has a
smooth dependence on a parameter g: H共g兲. The groundstate average of a local operator O of the system, 具O典共g兲,
naturally also depend on g. If the function 具O典共g兲, in the
limit of infinite system size, has a singularity at gc for some
local operators O, then the system described by H共g兲 has a
quantum phase transition at gc. After defining phase transition, we can define when two quantum ground states belong
to the same phase: let 兩⌽共0兲典 be the ground state of H共0兲 and
兩⌽共1兲典 be the ground state of H共1兲. If we can find a smooth
path H共g兲, 0 ⱕ g ⱕ 1 that connect the two Hamiltonian H共0兲
and H共1兲 such that there is no phase transition along the
path, then the two quantum ground states 兩⌽共0兲典 and 兩⌽共1兲典
belong to the same phase. We note that “connected by a
smooth path” define an equivalence relation between quantum states. A quantum phase is an equivalence class of such
equivalence relation. Such an equivalence class is called an
universality class.
If 兩⌽共0兲典 is the ground state of H共0兲 and in the limit of
infinite system size all excitations above 兩⌽共0兲典 have a gap,
then for small enough g, we believe that the systems described by H共g兲 are also gapped.34 In this case, we can show
that, the ground state of H共g兲, 兩⌽共g兲典, is in the same phase as
兩⌽共0兲典, for small enough g, i.e., the average 具O典共g兲 is a
E
E
∆
∆
ε
(a)
s
E
(b)
s
(d)
s
E
(c)
s
FIG. 1. 共Color online兲 Energy spectrum of a gapped system as a
function of a parameter s in the Hamiltonian. 关共a兲 and 共b兲兴 For
gapped system, a quantum phase transition can happen only when
energy gap closes. 共a兲 describes a first-order quantum phase transition 共caused by level crossing兲. 共b兲 describes a continuous quantum
phase transition which has a continuum of gapless excitations at the
transition point. 共c兲 and 共d兲 cannot happen for generic states. A
gapped system may have ground-state degeneracy, where the energy splitting between the ground states vanishes when system size
L → ⬁: limL→⬁ ⑀ = 0. The energy gap ⌬ between ground and excited
states on the other hand remains finite as L → ⬁.
smooth function of g near g = 0 for any local operator O.35
After scaling g to 1, we find that: if the energy gap for H共g兲
is finite for all g in the segment 关0,1兴, then there is no phase
transition along the path g.
In other words, for gapped system, a quantum phase transition can happen only when energy gap closes35 共see Fig.
1兲.36 Here, we would like to assume that the reverse is also
true: a closing of the energy gap for a gapped state always
induces a phase transition. Or more precisely if two gapped
states 兩⌽共0兲典 and 兩⌽共1兲典 are in the same phase, then we can
always find a family of Hamiltonian H共g兲 , such that the
energy gap for H共g兲 are finite for all g in the segment [0,1],
and 兩⌽共0兲典 and 兩⌽共1兲典 are ground states of H共0兲 and H共1兲 ,
respectively.
The above two statements imply that two gapped quantum
states are in the same phase 兩⌽共0兲典 ⬃ 兩⌽共1兲典 if and only if
they can be connected by an adiabatic evolution that does not
close the energy gap.
Given two states, 兩⌽共0兲典 and 兩⌽共1兲典, determining the existence of such a gapped adiabatic connection can be hard.
We would like to have a more operationally practical equivalence relation between states in the same phase. Here we
would like to show that two gapped states 兩⌽共0兲典 and 兩⌽共1兲典
are in the same phase, if and only if they are related by a
local unitary 共LU兲 evolution. We define a LU evolution as a
unitary operation generated by time evolution of a local
Hamiltonian for a finite time. That is,
兩⌽共1兲典 ⬃ 兩⌽共0兲典
iff 兩⌽共1兲典 = T关e−i兰0dgH共g兲兴兩⌽共0兲典, 共1兲
1
˜
where T is the path-ordering operator and H̃共g兲 = 兺iOi共g兲 is a
sum of local Hermitian operators. Note that H̃共g兲 is in general different from the adiabatic path H共g兲 that connects the
two states.
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CHEN, GU, AND WEN
First, assume that two states 兩⌽共0兲典 and 兩⌽共1兲典 are in the
same phase, therefore we can find a gapped adiabatic path
H共g兲 between the states. The existence of a gap prevents the
system to be excited to higher energy levels and leads to a
local unitary evolution, the quasiadiabatic continuation as
defined in Ref. 35, that maps from one state to the other. That
is,
兩⌽共1兲典 = U兩⌽共0兲典,
U = T关e−i兰0dgH共g兲兴 .
1
˜
共2兲
The exact form of H̃共g兲 is given in Refs. 34 and 35 and will
be discussed in more detail in the Appendix.
On the other hand, the reverse is also true: if two gapped
states 兩⌽共0兲典 and 兩⌽共1兲典 are related by a local unitary evolution, then they are in the same phase. Since 兩⌽共0兲典 and
兩⌽共1兲典 are related by a local unitary evolution, we have
1 ˜
兩⌽共1兲典 = T关e−i兰0dgH共g兲兴兩⌽共0兲典. Let us introduce
兩⌽共s兲典 = U共s兲兩⌽共0兲典,
U共s兲 = T关e−i兰0dgH共g兲兴 .
s
˜
共3兲
Assume 兩⌽共0兲典 is a ground state of H共0兲, then 兩⌽共s兲典 is a
ground state of H共s兲 = U共s兲HU†共s兲. If H共s兲 remains local and
gapped for all s 苸 关0 , 1兴, then we have found an adiabatic
connection between 兩⌽共0兲典 and 兩⌽共1兲典.
First, let us show that H共s兲 is a local Hamiltonian. Since
H is a local Hamiltonian, it has a form H = 兺iOi, where Oi
only acts on a cluster whose size is ␰. ␰ is called the range of
interaction of H. We see that H共s兲 has a form
H共s兲 = 兺iOi共s兲, where Oi共s兲 = U共s兲OiU†共s兲. To show that
Oi共s兲 only acts on a cluster of a finite size, we note that for a
local system described by H̃共g兲, the propagation velocities of
its excitations have a maximum value vmax. Since Oi共s兲 can
be viewed as the time evolution of Oi by H̃共t兲 from t = 0 to
t = s, we find that Oi共s兲 only acts on a cluster of size
␰ +˜␰ + svmax,35,37 where ˜␰ is the range of interaction of H̃.
Thus H共s兲 are indeed local Hamiltonian.
If H has a finite energy gap, then H共s兲 also have a finite
energy gap for any s. As s goes for 0 to 1, the ground state of
the local Hamiltonians, H共s兲, goes from 兩⌽共0兲典 to 兩⌽共1兲典.
Thus the two states 兩⌽共0兲典 and 兩⌽共1兲典 belong to the same
phase. This completes our argument that states related by a
local unitary evolution belong to the same phase.
The finiteness of the evolution time is very important in
the above discussion. Here “finite” means the evolution time
does not grow with system size and in the thermodynamic
limit, phases remain separate under such evolutions, as
proven in Ref. 38. On the other hand, if the system size
under consideration is finite, there is a critical time limit
above which phase separation could be destroyed. The time
limit depends on the propagation speed of interactions in the
Hamiltonian. This is the case in Ref. 39, where topological
order as measured by topological entropy and fidelity was
found to decay under certain local Hamiltonian evolution 共a
quantum quench兲. However this result does not contradict
our statement. As the calculation is done for a particular
system size, the critical time limit could be below or above
the time period they studied. If the calculation could be done
for larger and larger system sizes for a fixed amount of time,
we expect that topological order should emerge as stable
against local quenches.
Thus through the above discussion, we show that two
gapped ground states,40 兩⌽共0兲典 and 兩⌽共1兲典 , belong to the
same phase if and only if they are related by a local unitary
evolution [Eq. (1)]. A more detailed and more rigorous discussion of this equivalence relation is given in the Appendix,
where exact bounds on locality and transformation error is
given.
The relation 关Eq. 共1兲兴 defines an equivalence relation between 兩⌽共0兲典 and 兩⌽共1兲典. The equivalence classes of such an
equivalence relation represent different quantum phases. So
the above result implies that the equivalence classes of the
LU evolutions are the universality classes of quantum phases
for gapped states.
IV. TOPOLOGICAL ORDER IS A PATTERN
OF LONG-RANGE ENTANGLEMENT
Using the LU evolution, we can obtain a more precise
description 共or definition兲 of short-range entanglement: a
state has only short-range entanglement if and only if it can
be transformed into an unentangled state (i.e., a directproduct state) through a local unitary evolution.
If a state cannot be transformed into an unentangled state
through a LU evolution, then the state has long-range entanglement 共LRE兲. We also see that all states with shortrange entanglement can transform into each other through
local unitary evolutions.
Thus all states with short-range entanglement belong to
the same phase. The local unitary evolutions we consider
here do not have any symmetry. If we require certain symmetry of the local unitary evolutions, states with short-range
entanglement may belong to different symmetry breaking
phases, which will be discussed in Sec. VI.
Since a direct-product state is a state with trivial topological order, we see that a state with a short-range entanglement
also has a trivial topological order. This leads us to conclude
that a nontrivial topological order is related to long-range
entanglement. Since two gapped states related by a LU evolution belong to the same phase, thus two gapped states related by a local unitary evolution have the same topological
order. In other words, topological order describes the
equivalent classes defined by local unitary evolutions.
Or more pictorially, topological order is a pattern of longrange entanglement. In Ref. 38, it was shown that the “topologically nontrivial” ground states, such as the toric code,27
cannot be changed into a “topologically trivial” state such as
a product state by any unitary locality-preserving operator. In
other words, those topologically nontrivial ground states
have long-range entanglement.
V. LU EVOLUTIONS AND QUANTUM CIRCUITS
The LU evolutions introduced here
quantum circuits with finite depth. To
cuits, let us introduce piecewise local
piecewise local unitary operator has
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is closely related to
define quantum cirunitary operators. A
a form U pwl = 兿iUi,
PHYSICAL REVIEW B 82, 155138 共2010兲
LOCAL UNITARY TRANSFORMATION, LONG-RANGE…
g2
g2
SRE
1 2 ... l
Ui
(b)
LRE 1
(a)
FIG. 2. 共Color online兲 共a兲 A graphic representation of a quantum
circuit, which is formed by 共b兲 unitary operations on patches of
finite size l. The dark shading represents a causal structure.
where 兵Ui其 is a set of unitary operators that act on nonoverlapping regions. The size of each region is less than some
finite number l. The unitary operator U pwl defined in this way
is called a piecewise local unitary operator with range
l. A quantum circuit with depth M is given by the product
of M piecewise local unitary operators 共see Fig. 2兲:
共2兲
共M兲
M
= U共1兲
Ucirc
pwlU pwl ¯ U pwl . In quantum information theory, it is
known that finite time unitary evolution with local Hamiltonian 共LU evolution defined before兲 can be simulated with
constant depth quantum circuit and vice versa. Therefore, the
equivalence relation 关Eq. 共1兲兴 can be equivalently stated in
terms of constant depth quantum circuits,
兩⌽共1兲典 ⬃ 兩⌽共0兲典
M
iff 兩⌽共1兲典 = Ucirc
兩⌽共0兲典,
共4兲
where M is a constant independent of system size. Because
of their equivalence, we will use the term “local unitary
transformation” to refer to both local unitary evolution and
constant depth quantum circuit in general.
The LU transformation defined through LU evolution
关Eq. 共1兲兴 is more general. It can be easily generalized to
study topological orders and quantum phases with symmetries 共see Sec. VI兲.30,41 The quantum circuit has a more clear
and simple causal structure. However, the quantum circuit
approach breaks the translation symmetry. So it is more suitable for studying quantum phases that do not have translation
symmetry.
In fact, people have been using quantum circuits to classify many-body quantum states which correspond to quantum phases of matter. In Ref. 42, the local unitary transformations described by quantum circuits was used to define a
renormalization-group 共RG兲 transformations for states and
establish an equivalence relation in which states are equivalent if they are connected by a local unitary transformation.
Such an approach was used to classify one-dimensional matrix product states. In Ref. 43, the local unitary transformations with disentanglers was used to perform a
renormalization-group transformations for states, which give
rise to the multiscale entanglement renormalization ansatz
共MERA兲 in one and higher dimensions. The disentanglers
and the isometries in MERA can be used to study quantum
phases and quantum phase transitions in one and higher dimensions. For a class of exactly solvable Hamiltonians
which come from the stabilizer codes in quantum computation, topological order has also been classified using local
unitary circuits.44 Later in this paper, we will use the quantum circuit description of LU transformations to classify
two-dimensional 共2D兲 topological orders through classifying
the fixed-point LU transformations.
SB−SRE 1
SB−SRE 2
SY−SRE 1
SY−SRE 2
SY−LRE 1 SY−LRE 2 SY−LRE 3
LRE 2
SB−LRE 1 SB−LRE 2 SB−LRE 3
g1
(a)
(b)
g1
FIG. 3. 共Color online兲 共a兲 The possible phases for a Hamiltonian
H共g1 , g2兲 without any symmetry. 共b兲 The possible phases for a
Hamiltonian Hsymm共g1 , g2兲 with some symmetries. The shaded regions in 共a兲 and 共b兲 represent the phases with short-range entanglement 共i.e., those ground states can be transformed into a direct
product state via a generic LU transformations that do not have any
symmetry兲.
VI. SYMMETRY-BREAKING ORDERS AND SYMMETRY
PROTECTED TOPOLOGICAL ORDERS
In the above discussions, we have defined phases without
any symmetry consideration. The H̃共g兲 or U pwl in the LU
transformation does not need to have any symmetry and can
be sum/product of any local operators. In this case, two
Hamiltonians with an adiabatic connection are in the same
phase even if they may have different symmetries. Also, all
states with short-range entanglement belong to the same
phase 共under the LU transformations that do not have any
symmetry兲.
On the other hand, we can consider only Hamiltonians H
with certain symmetries and define phases as the equivalent
classes of symmetric local unitary transformations,
兩⌿典 ⬃ T共e−i兰0dgH共g兲兲兩⌿典
1
˜
or
M
兩⌿典 ⬃ Ucirc
兩⌿典,
M
has the same symmetries as H. We note
where H̃共g兲 or Ucirc
that the symmetric local unitary transformation in the form
1 ˜
T共e−i兰0dgH共g兲兲 always connect to the identity transformation
continuously. This may not be the case for the transformation
M
. To rule out that possibility, we define symin the form Ucirc
metric local unitary transformations as those that connect to
the identity transformation continuously.
The equivalent classes of the symmetric LU transformations have very different structures compared to those of LU
transformations without symmetry. Each equivalent class of
the symmetric LU transformations is smaller and there are
more kinds of classes, in general.
Figure 3 compares the structure of phases for a system
without any symmetry and a system with some symmetry in
more detail. For a system without any symmetry, all the SRE
states 共i.e., those ground states can be transformed into a
direct product state via a generic LU transformations that do
not have any symmetry兲 are in the same phase 关SRE in Fig.
3共a兲兴. On the other hand, LRE can have many different patterns that give rise to different topological phases 关LRE 1
and LRE 2 in Fig. 3共a兲兴. The different topological orders
usually give rise to quasiparticles with different fractional
statistics and fractional charges.
For a system with some symmetries, the phase structure
can be much more complicated. The short-range-entangled
states no longer belong to the same phase since the equiva-
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CHEN, GU, AND WEN
lence relation is described by more special symmetric LU
transformations: 共A兲 states with short-range entanglement
belong to different equivalent classes of the symmetric LU
transformations if they have different broken symmetries.
They correspond to the SB short-range-entanglement phases
SB-SRE 1 and SB-SRE 2 in Fig. 3共b兲. They are Landau’s
symmetry-breaking states.
共B兲 States with short-range entanglement can belong to
different equivalent classes of the symmetric LU transformations even if they do not break any symmetry of the system.
共In this case, they have the same symmetry.兲 They correspond to the symmetric 共SY兲 short-range-entangled phases
SY-SRE 1 and SY-SRE 2 in Fig. 3共b兲. We say those states
have symmetry protected topological orders. Haldane
phase45 and Sz = 0 phase of spin-1 chain are examples of
states with the same symmetry which belong to two different
equivalent classes of symmetric LU transformations 共with
parity symmetry兲.41,46 Band and topological insulators47–52
are other examples of states that have the same symmetry
and at the same time belong to two different equivalent
classes of symmetric LU transformations 共with time reversal
symmetry兲. Also, for a system with some symmetries, the
long-range-entangled states are divided into more classes
共more phases兲.
共C兲 Symmetry-breaking and long-range entanglements
can appear together in a state, such as SB-LRE 1, SB-LRE 2,
etc., in Fig. 3共b兲. The topological superconducting states are
examples of such phases.53,54
共D兲 Long-range-entangled states that do not break any
symmetry can also belong to different phases such as the
symmetric long-range-entanglement phases SY-LRE 1, SYLRE 2, etc., in Fig. 3共b兲. The many different Z2 symmetric
spin liquids with spin rotation, translation, and time-reversal
symmetries are examples of those phases.30,55,56 Some timereversal symmetric topological orders were also called topological Mott insulators or fractionalized topological
insulators.57–62
VII. LOCAL UNITARY TRANSFORMATION AND WAVE
FUNCTION RENORMALIZATION
After defining topological order as the equivalent classes
of many-body wave functions under LU transformations, we
like to ask: how to describe 共or label兲 the different equivalent
classes 共i.e., the different topological orders or patterns of
long-range entanglement兲?
One simple way to do so is to use the full wave function
which completely describe the different topological orders.
But the full wave functions contain a lot of nonuniversal
short-range entanglement. As a result, such a labeling
scheme is a very inefficient many-to-one labeling scheme of
topological orders. To find a more efficient or even one-toone labeling scheme, we need to remove the nonuniversal
short-range entanglement.
As the first application of the notion of LU transformation, we would like to describe a wave function
renormalization-group flow introduced in Refs. 43 and 63.
The wave function renormalization can remove the shortrange entanglement and simplify the wave function. In Ref.
FIG. 4. 共Color online兲 A finite depth quantum circuit can transform a state 兩⌽典 into a direct-product state, if and only if the state
兩⌽典 has no long-range quantum entanglement. Here, a dot represents a site with physical degrees of freedom. A vertical line carries
an index that label the different physical states on a site. The presence of horizontal lines between dots represents quantum
entanglement.
63, the wave function renormalization for string-net states is
generated by the following two basic moves:
k
i
Φ
=δ
Φ
ij
i l j
共5兲
Φ
i
j
m
l
k
=
n
∗
jim
Flk
∗ n∗ Φ
i
l
j n k
共6兲
共Note that the definition of the F-tensor in Ref. 63 is slightly
different from the definition in this paper.兲 The two basic
moves can generate a generic wave function renormalization
which can reduce the string-net wave functions to very
simple forms.17,63 Later in Ref. 43, the wave function renormalization for generic states was discussed in a more general
setting, and was called MERA. The two basic string-net
moves 关Eqs. 共5兲 and 共6兲兴 correspond to the isometry and the
disentangler in MERA, respectively. In the MERA approach,
the isometries and the disentanglers are applied in a layered
fashion while in the string-net approach, the two basic moves
can be applied arbitrarily. In this section, we will follow the
MERA setup to describe the wave function renormalization.
Later in this paper, we will follow the string-net setup to
study the fixed-point wave functions.
Note that we can use a LU transformation U to transform
some degrees of freedom in a state into direct product 共see
Figs. 4 and 5兲. We then remove those degrees of freedom in
the form of direct product. Such a procedure does not change
the topological order. The reverse process of adding degrees
of freedom in the form of direct product also does not change
the topological order. We call the local transformation in Fig.
5 that changes the degrees of freedom a generalized local
FIG. 5. 共Color online兲 A piecewise local unitary transformation
can transform some degrees of freedom in a state ⌽ into a direct
product. Removing/adding the degrees of freedom in the form of
direct product defines an additional equivalence relation that defines
the topological order 共or classes of long-range entanglement兲.
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VIII. WAVE FUNCTION RENORMALIZATION AND A
CLASSIFICATION OF TOPOLOGICAL ORDER
U
region A
(a)
U
P
U
region A
(b)
FIG. 6. 共Color online兲 共a兲 A gLU transformation U acts in region
A of a state 兩⌽典, which reduces the degree of freedom in region A to
those contained only in the support space of 兩⌽典 in region A. 共b兲
U†U = P is a projector that does not change the state 兩⌽典.
unitary 共gLU兲 transformation. It is clear that a generalized
local unitary transformation inside a region A does not
change the reduced density matrix ␳A for the region A. This
is the reason why we say that 共generalized兲 local unitary
transformations cannot change long-range entanglement and
topological order. Similarly, the addition or removal of decoupled degrees of freedom to or from the Hamiltonian,
H ↔ H 丢 Hdp, will not change the phase of the Hamiltonian
共i.e., the ground states of H and H 丢 Hdp are in the same
phase兲, if those degrees of freedom form a direct product
state 共i.e., the ground state of Hdp is a direct product state兲.
Let us define the gLU transformation U more carefully
and in a more general setting. Consider a state 兩⌽典. Let ␳A be
the reduced density matrix of 兩⌽典 in region A. ␳A may act in
a subspace of the total Hilbert space VA in region A, which is
called the support space VAsp of region A. The dimension DAsp
of VAsp is called support dimension of region A. Now the
Hilbert space VA in region A can be written as VA = VAsp
sp
丣 V̄A . Let 兩˜
␺i典, i = 1 , . . . , DAsp be a basis of this support space
VAsp, 兩˜␺i典, i = DAsp + 1 , . . . , DA be a basis of V̄Asp, where DA is the
dimension of VA, and 兩␺i典, i = 1 , . . . , DA be a basis of VA. We
can introduce a LU transformation U full which rotates the
basis 兩␺i典 to 兩˜␺i典. We note that in the new basis, the wave
function only has nonzero amplitudes on the first DAsp basis
vectors. Thus, in the new basis 兩˜␺i典, we can reduce the range
of the label i from 关1 , DA兴 to 关1 , DAsp兴 without losing any
information. This motivates us to introduce the gLU transformation as a rotation from 兩␺i典, i = 1 , . . . , DA to 兩˜␺i典, i
= 1 , . . . , DAsp. The rectangular matrix U is given by Uij
= 具˜␺i 兩 ␺ j典. We also regard the inverse of U , U†, as a gLU
transformation. A LU transformation is viewed as a special
case of gLU transformation where the degrees of freedom
are not changed. Clearly U†U = P and UU† = P⬘ are two projectors. The action of P does not change the state 兩⌽典 关see
Fig. 6共b兲兴.
We note that despite the reduction in the degrees of freedom, a gLU transformation defines an equivalent relation.
Two states related by a gLU transformation belong to the
same phase. The renormalization flow induced by the gLU
transformations always flows within the same phase.
As an application of the wave function renormalization, in
this section, we will study the structure of fixed-point wave
functions under the wave function renormalization, which
will lead to a classification of topological order 共without any
symmetry兲.
We note that as wave functions flow to a fixed point, the
gLU transformations in each step of the renormalization also
flow to a fixed point. So instead of studying fixed-point wave
functions, here, we will study the fixed-point gLU transformations. For this purpose, we need to fix the renormalization
scheme. In the following, we will discuss a renormalization
scheme motivated by the string-net wave function.63 After
we specify a proper wave function renormalization scheme,
then the fixed-point wave function is simply the wave function whose “form” does not change under the wave function
renormalization.
Since those fixed-point gLU transformations do not
change the fixed-point wave function, their actions on the
fixed-point wave function do not depend on the order of the
actions. This allows us to obtain many conditions that gLU
transformations must satisfy. From those conditions, we can
determine the forms of allowed fixed-point gLU transformations. This leads to a general description and a classification
of topological orders and their corresponding fixed-point
wave functions.
The renormalization scheme that we will discuss was first
used in Ref. 63 to characterize the scale invariant string-net
wave function. It is also used in Ref. 17 to simplify the
string-net state in a region, which allows us to calculate the
entanglement entropy of the string-net state exactly. A similar approach was used in Ref. 64 to show quantum-double/
string-net wave function to be a fixed-point wave function
and its connection to 2D MERA.43 In the following, we will
generalize those discussions by not starting with string-net
wave functions. We just try to construct local unitary transformations at a fixed point. We will see that the fixed-point
conditions on the gLU transformations lead to a mathematical structure that is similar to the tensor category theory—the
mathematical framework behind the string-net states.
A. Quantum states on a graph
Since the wave function renormalization may change the
lattice structure, we will consider quantum states defined on
a generic trivalence graph G: each edge has N + 1 states,
labeled by i = 0 , . . . , N 共see Fig. 7兲. We assume that the index
i on the edge admits an one-to-one mapping ⴱ: i → iⴱ that
satisfies 共iⴱ兲ⴱ = i. As a result, the edges of the graph are oriented. The mapping i → iⴱ corresponds to the reverse of the
orientation of the edge 共see Fig. 8兲. Each vertex also has
physical states, labeled by ␣ = 1 , . . . , Nv 共see Fig. 7兲.
Each labeled graph 共see Fig. 7兲 corresponds to a state and
all the labeled graphs form an orthonormal basis. Our fixedpoint state is a superposition of those basis states:
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CHEN, GU, AND WEN
m
γ
i=0,...,N
m
l
β
k
j
α= 1,...,N v
γ
l
β
k
i
j
λ
α= 1,...,Νijk
λ
n
n
FIG. 7. A quantum state on a graph G. There are N + 1 states on
each edge which are labeled by l = 0 , . . . , N. There are Nv states on
each vertex which are labeled by ␣ = 1 , . . . , Nv.
FIG. 9. A quantum state on a graph G. If the three edges of a
vertex are in the states i, j, and k, respectively, then the vertex has
Nijk states, labeled by ␣ = 1 , . . . , Nijk. Note the orientation of the
edges are point toward to vertex. Also note that i → j → k runs
anticlockwise.
|Φfix =
all conf.
Φfix
.
i
Here we will make an important assumption about the
fixed-point wave function. We will assume that the fixedpoint wave function is “topological:” two labeled graphs
have the same amplitude if the two labeled graphs can be
deformed into each other continuously on the plane without
the vertices crossing the links. For example,
⎛
⎞
⎛
⎞
ψfix ⎝1
0
1⎠
2 1
0
1
= ψfix ⎝2
2 1
0
1
0⎠
0
α
m
j
k
β
.
l
The fixed-point wave function
i
Φ fix
j
α
m
k
β
l
共only the relevant part of the graph is drawn兲 can be viewed
as a function of ␣ , ␤ , m:
i j k
ψijkl,Γ (α, β, m) = Φfix
Due to such an assumption, the topological orders studied in
this paper may not be most general.
We also assume that all the fixed-point states on each
different graphs to have the same form. This assumption is
motivated by the fact that during wave function renormalization, we transform a state on one graph to a state on a different graph. The “fixed point” means that the wave functions on those different graphs are all determined by the
same collection of the rules, which defines the meaning of
having the same form. However, the wave function for a
given graph can have different total phases if the wave function is calculated by applying the rules in different orders.
Those rules are noting but the fixed-point gLU transformations.
B. Structure of entanglement in a fixed-point wave function
i
i*
FIG. 8. The mapping i → iⴱ corresponds to the reverse of the
orientation of the edge.
m
β
l
if we fix i , j , k , l and the indices on other part of the graph.
共Here the indices on other part of the graph is summarized by
⌫.兲 As we vary the indices ⌫ on other part of graph 共still
keeping i, j, k, and l fixed兲, the wave function of ␣ , ␤ , m,
␺ijkl,⌫共␣ , ␤ , m兲, may change. All those ␺ijkl,⌫共␣ , ␤ , m兲 form a
linear space of dimension Dijklⴱ. Dijkl is an important concept
that will appear later. We note that the two vertices ␣ and ␤
and the link m form a region surrounded by the links i , j , k , l.
So we will call the dimension-Dijklⴱ space the support space
Vijklⴱ and Dijklⴱ the support dimension for the state ⌽fix on the
region surrounded by the fixed boundary state i , j , k , l.
Similarly, we can define Dijk as the support dimension of
the
Φ fix
Before describing the wave function renormalization, let
us examine the structure of entanglement of a fixed-point
wave function. First, let us consider a fixed-point wave function ⌽fix on a graph. We examine the wave function on a
patch of the graph, for example,
α
j
i
α
k
on a region bounded by links i , j , k. Since the region contains
only a single vertex ␣ with Nv physics states, we have
Dijk ⱕ Nv. We can use a local unitary transformation on the
vertex to reduce the range of ␣ to 1 , . . . , Nijk, where
Nijk = Dijk. In the rest of this paper, we will implement such a
reduction. So, the number of physical states on a vertex depend on the physical states of the edges that connect to the
vertex. If the three edges of a vertex are in the states i, j, and
k, respectively, then the vertex has Nijk states, labeled by
␣ = 1 , . . . , Nijk 共see Fig. 9兲. Here we assume that
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We note that in the fixed-point wave function
i j k
α
Φfix
m
β
Again, such a wave function can be viewed as a function of
␹ , ␦ , n:
i j k
,
l
N
the number of choices of ␣ , ␤ , m is Nijklⴱ = 兺m=0
N jimⴱNkmlⴱ.
Thus the support dimension Dijklⴱ satisfies Dijklⴱ ⱕ Nijklⴱ.
Here we will make an important assumption—the saturation
assumption: the fixed-point wave function saturate the inequality,
if we fix i , j , k , l and the indices on other part of the graph.
The support dimension of the state
i j k
χ
δ n
l
Φfix
N
兺 N jim Nkml .
Dijklⴱ = Nijklⴱ ⬅
ⴱ
共8兲
ⴱ
m=0
We will see that the entanglement structure described by
such a saturation assumption is invariant under the wave
function renormalization.
C. First type of wave function renormalization
Our wave function renormalization scheme contains two
types of renormalization. The first type of renormalization
does not change the degrees of freedom and corresponds to a
local unitary transformation. It corresponds to locally deform
the graph
i
α
m
j
α
Φfix
and
m
i
Φfix
β
l
j χk
δ n
l
β
are related via a local unitary transformation. Thus Dijklⴱ
= D̃ijklⴱ, which implies
l
N
j χk
δ n .
l
χ
δ n
l
N
兺 N jim Nkml = n=0
兺 Nkjn Nl ni .
ⴱ
ⴱ
ⴱ
共9兲
ⴱ
m=0
共The parts that are not drawn are the same.兲 The fixed-point
wave function on the new graph is given by
i j k
Φ fix
on the region surrounded by i , j , k , l is D̃ijklⴱ. Again
N
D̃ijklⴱ ⱕ Ñijklⴱ, where Ñijklⴱ ⬅ 兺n=0
NkjnⴱNlⴱni is number of
choices of ␹ , ␦ , n. The saturation assumption implies that
Ñijklⴱ = D̃ijklⴱ.
The two fixed-point wave functions
i j k
k
to
i
χ
δ n
l
ψ̃ijkl,Γ (χ, δ, n) = Φfix
We express such an unitary transformation as
N Nkjnⴱ Nnilⴱ
ijm,␣␤˜
Fkln,
兺
兺
␹␦ ␺ijkl,⌫共␹, ␦,n兲
n=0 ␹=1 ␦=1
␺ijkl,⌫共␣, ␤,m兲 ⯝ 兺
.
共10兲
or graphically as
⎛
Φfix ⎝
i
α
m
j
k
β
l
⎞
⎠
kjn∗ Nnil∗
N N
=0
=1
where ⯝ means equal up to a constant phase factor. 共Note
that the total phase of the wave function is unphysical.兲 We
will call such a wave function renormalization step a
F-move.
We would like to remark that Eq. 共11兲 relates two wave
functions on two graphs G1 and G2 which only differ by a
ijm,␣␤
local reconnection. We can choose the phase of Fkln,
␹␦ to
make ⯝ into =:
⎛
ijm,αβ
Fkln,χδ
Φfix
⎝
i
δ=1
⎛
Φfix ⎝
i
α
m
j
k
β
l
⎞
⎠=
⎞
j χk
δ n ⎠.
l
共11兲
kjn∗ Nnil∗
N N
n=0 χ=1
δ=1
⎛
ijm,αβ
Fkln,χδ
Φfix
⎝
i
⎞
j χk
δ n ⎠.
l
But such choice of phase only works for a particular pair of
ijm,␣␤
graphs G1 and G2. To use Fkln,
␹␦ to relate all pair of states
that only differ by a local reconnection, in general, we may
have a phase ambiguity, with the value of phase depend on
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PHYSICAL REVIEW B 82, 155138 共2010兲
CHEN, GU, AND WEN
the pair of graphs. So Eq. 共11兲 can only be a relation up to a
total phase factor.
For fixed i, j, k, and l, the matrix Fijkl with matrix elements
ijm,␣␤
ij m,␣␤
共Fkl兲n,␹␦ = Fkln,
␹␦ is a Nijklⴱ dimensional matrix 关see Eq. 共9兲兴.
The mapping ˜␺ijkl,⌫共␹ , ␦ , n兲 → ␺ijkl,⌫共␣ , ␤ , m兲 generated by
the matrix Fijkl is unitary. Since, as we change ⌫,
˜␺
ijkl,⌫共␹ , ␦ , n兲 and ␺ijkl,⌫共␣ , ␤ , m兲 span two Nijklⴱ-dimensional
spaces. Thus Fijkl is a Nijklⴱ ⫻ Nijklⴱ unitary matrix
ijm⬘,␣⬘␤⬘ ijm,␣␤ ⴱ
Fkln,
共Fkln,␹␦ 兲 = ␦m␣␤,m⬘␣⬘␤⬘ ,
兺
␹␦
n,␹,␦
共12兲
⎛
Φfix ⎝
i
⎞
j χk
δ n ⎠
l
α
m
j
k
k
β
to
β
m
l
l
i
α
⎝
Fljkn,χδ
∗ i∗ m∗ ,βα Φfix
m=0 α,β
jkn,␹␦
ijm,␣␤ ⴱ
共Fkln,
␹␦ 兲 ⯝ Flⴱiⴱmⴱ,␤␣ .
i
α
m
j
k
⎞
β
⎠
l
共16兲
共17兲
Since the total phase of the wave function is unphysical, the
ijm,␣␤
total phase of Fkln,
␹␦ can be chosen arbitrarily. We can
ijm,␣␤
choose the total phase of Fkln,
␹␦ to make
jkn,␹␦
ijm,␣␤ ⴱ
共Fkln,
␹␦ 兲 = Flⴱiⴱmⴱ,␤␣ .
共18兲
If we apply Eq. 共18兲 twice, we reproduce Eq. 共14兲. Thus Eq.
共14兲 is not independent and can be dropped.
From the graphic representation 关Eq. 共11兲兴, We note that
,
j
ijm,␣␤
Fkln,
␹␦ = 0
and
i
We see that
⎛
k
Φfix ⎝
β
m
j χk
δ n
l
k
to
l
δ
χn
j
i
.
when
N jimⴱ ⬍ 1
l
i
α
j
⎞
⎠
N ⎛
∗
∗
,βα
Fijkl∗ nm∗ ,δχ
Φfix ⎝
n=0 χ,δ
k
l
δ
χn
j
i
klⴱmⴱ,␤␣
ijm,␣␤
Fkln,
␹␦ ⯝ Fijⴱnⴱ,␦␹
⎠.
共13兲
共14兲
共where we have used the condition Nijkl = Dijkl.兲
Using the relation 关Eq. 共12兲兴, we can rewrite Eq. 共11兲 as
⎛
⎞
⎛
⎞
i j χk
i j
k
N α
ijm,αβ ∗
Φfix ⎝ δ n ⎠ (Fkln,χδ
) Φfix ⎝ m β ⎠ .
l
l
m=0 α,β
or
Nkmlⴱ ⬍ 1
共15兲
Φfix
j χ
δ n
l
Nnilⴱ ⬍ 1.
When N jimⴱ ⬍ 1 or Nkmlⴱ ⬍ 1, the left-hand side of Eq. 共11兲 is
ijm,␣␤
always zero. Thus Fkln,
␹␦ = 0 when N jimⴱ ⬍ 1 or Nkmlⴱ ⬍ 1.
When Nkjnⴱ ⬍ 1 or Nnilⴱ ⬍ 1, wave function on the right-hand
side of Eq. 共11兲 is always zero. So we can choose
ijm,␣␤
Fkln,
␹␦ = 0 when Nkjnⴱ ⬍ 1 or Nnilⴱ ⬍ 1.
The F-move 关Eq. 共11兲兴 maps the wave functions on two
different graphs through a local unitary transformation. Since
we can locally transform one graph to another graph through
different paths, the F-move 关Eq. 共11兲兴 must satisfy certain
self-consistent condition. For example, the graph
i
α
m
j
β
n
k
l
χ
p
can be transformed to
j
k
γ
i
or
共19兲
i
We can also express
Nkjnⴱ ⬍ 1
or
⎞
Equations 共11兲 and 共13兲 relate the same pair of graphs, and
thus
as
⎛
using the relabeled Eq. 共11兲. So we have
where ␦m␣␤,m⬘␣⬘␤⬘ = 1 when m = m⬘, ␣ = ␣⬘, ␤ = ␤⬘, and
␦m␣␤,m⬘␣⬘␤⬘ = 0 otherwise.
We can deform
i
N k
φ
s
δ
q
l
p
through two different paths; one contains two steps of local
transformations and the other contains three steps of local
transformations as described by Eq. 共11兲. The two paths lead
to the following relations between the wave functions:
155138-10
PHYSICAL REVIEW B 82, 155138 共2010兲
LOCAL UNITARY TRANSFORMATION, LONG-RANGE…
⎛
⎜
Φfix ⎜
⎝
i
α
m
j
k
β
l
χ
n
⎞
⎛
⎟ mkn,βχ
⎜
⎟
Flpq,δ Φfix ⎜
⎠
⎝
⎜
Φfix ⎜
⎝
i
α
m
j
k
β
n
j
α
k
m
ε
q,δ,
p
⎛
i
l
χ
⎞
⎛
⎟ ijm,αβ
⎜
⎟
Fknt,ηϕ Φfix ⎜
⎠
⎝
t,η,ϕ
t,η,κ;ϕ;s,κ,γ;q,δ,φ
i
j
k
η
ϕt
n χ
p
⎛
⎟
⎟
⎠
p
⎞
p
δ
q
l
l
q,δ,;s,φ,γ
⎜
mkn,βχ ijm,α
Flpq,δ
Fqps,φγ Φfix ⎜
⎝
⎞
i
α
m
j
β
n
k
t,η,ϕ;s,κ,γ
⎜
ijm,αβ itn,ϕχ
Fknt,ηϕ
Flps,κγ Φfix ⎜
⎝
⎛
⎜
ijm,αβ itn,ϕχ jkt,ηκ
Fknt,ηϕ
Flps,κγ Flsq,δφ Φfix ⎜
⎝
i
j
⎟
⎠
⎛
α
⎜
ψ(α, β, χ, m, n) = Φfix ⎝
m
⎞
β
n
χ
⎟.
⎠
δ
φ
q
γ
s
p
l
i
j
⎞
s
δ
q
⎞
⎟
⎟,
⎠
共20兲
k
η
κ
t
γ s
p
l
⎞
⎟
⎟
⎠
⎟
⎟.
⎠
⎛
共21兲
i
j
⎜
Φfix ⎝
p
k
δ
φ
γ q
s
p
l
⎞
⎟
⎠
Nkjtⴱ Ntinⴱ Nltsⴱ
兺t ␩兺=1 ␸兺=1 ␬兺=1
As we vary indices on other part of graph, we obtain different wave functions ␺共␣ , ␤ , ␹ , m , n兲 which form a dimension
Dijklpⴱ space. In other words, Dijklpⴱ is the support dimension
of the state ⌽fix on the region ␣ , ␤ , ␹ , m , n with boundary
state i , j , k , l , p fixed 共see the discussion in Sec. VIII B兲.
Since the number of choices of ␣ , ␤ , ␹ , m , n is Nijklpⴱ
= 兺m,nN jimⴱNkmnⴱNlnpⴱ, we have Dijklpⴱ ⱕ Nijklpⴱ. Here we require a similar saturation condition as in Eq. 共8兲,
Nijklpⴱ = Dijklpⴱ .
φ
l
on the region bounded by i , j , k , l , p.
So the two relations 关Eqs. 共20兲 and 共21兲兴 can be viewed as
two relations between a pair of vectors in the two
Dijklpⴱ-dimensional vector spaces. As we vary indices on
other part of graph 共still keeping i , j , k , l , p fixed兲, each vector in the pair can span the full Dijklpⴱ-dimensional vector
space. So the validity of the two relations 关Eqs. 共20兲 and
共21兲兴 implies that
p
as a function of ␣ , ␤ , ␹ , m , n:
k
l
χ
γ
⎛
⎟
⎟
⎠
k
j
p
The consistence of the above two relations leads a condition
on the F tensor.
To obtain such a condition, let us fix i , j , k , l , p, and view
⎛
⎞
⎜
Φfix ⎝
i
共22兲
Nqmpⴱ
ijm,␣␤ itn,␸␹ jkt,␩␬
Fknt,
␩␸ Flps,␬␥Flsq,␦␾
=e
i␪F
兺
⑀=1
mkn,␤␹ ijm,␣⑀
Flpq,
␦⑀ Fqps,␾␥ ,
共23兲
which is a generalization of the famous pentagon identity
共due to the extra constant phase factor ei␪F兲. We will call such
a relation projective pentagon identity. The projective pentagon identity is a set of nonlinear equations satisfied by the
ijm,␣␤
rank-10 tensor Fkln,
␹␦ and ␪F. The above consistency relation
is equivalent to the requirement that the local unitary transformations described by Eq. 共11兲 on different paths all commute with each other up to a total phase factor.
D. Second type of wave function renormalization
Similarly, the number of choices of ␦ , ␾ , ␥ , q , s in
⎛
⎞
i
⎜
Φfix ⎝
k
j
γ
φ
s
δ
q
The second type of wave function renormalization does
change the degrees of freedom and corresponds to a generalized local unitary transformation. One way to implement
the second type renormalization is to reduce
l
⎟
⎠
p
j
is also Nijklpⴱ. Here we again assume D̃ijklpⴱ = Nijklpⴱ, where
D̃ijklpⴱ is the support dimension of
i
α
c
b
γ
β
a
i
to
k
j
λ
k
共the part of the graph that is not drawn is unchanged兲:
155138-11
PHYSICAL REVIEW B 82, 155138 共2010兲
CHEN, GU, AND WEN
k
k
j
i
i
⎛
k
j
i’
i
⎜
j
i’
Φfix ⎝
FIG. 10. A “triangle” graph can be transformed into a “tadpole”
via two steps of the first type of wave function renormalization 共i.e.,
two steps of local unitary transformations兲.
⎛
⎜
Φfix ⎝
j
i
α
c
b
γ
β
a
⎞
⎛
Nijk
⎟ abc,αβγ
⎜
Fijk,λ Φfix ⎝
⎠
λ=1
k
λ
k
j
This implies that
⎛
⎟
⎠.
⎜
Φfix ⎝
共24兲
k
j
j
β
⎞
⎟
⎜
⎠ δii Φfix ⎝
k i’
α
i
⎛
β
i
k i
α
兺 兺
兺
k,j ␣=1 ␤=1
␣␤
Pkj,
= 0,
i
k i’
α
共26兲
共27兲
␣␤ kj,␣␤ ⴱ
Pkj,
共Pi 兲 = 1
i
共28兲
if Nkiiⴱ ⬍ 1
or
N jⴱ jkⴱ ⬍ 1.
共29兲
The condition 关Eq. 共28兲兴 ensures that the two wave functions
on the two sides of Eq. 共27兲 have the same normalization. We
note that the number of choices for the four indices
␣␤
must be equal or greater than 1,
共j , k , ␣ , ␤兲 in Pkj,
i
to
Di = 兺 NiiⴱkN jkⴱ jⴱ ⱖ 1.
Notice that
⎞
⎛
β
i
j
k i
α
jj ∗ k,βα
⎟
Fi∗ i∗ m∗ ,λγ Φfix
⎠
m,λ,γ
i
j
γ m λ
i
⎛
m,λ,γ,l,ν,µ
⎛
⎜
Φfix ⎜
⎝
l
ν
⎜
∗
k,βα
i∗ mj,λγ
⎜
Fijj
∗ i∗ m∗ ,λγ Fm∗ i∗ l,νµ Φfix
⎝
⎞
µ
i
l
ν
i
m
⎟
⎟
⎠
共31兲
⎞
µ
i
共30兲
j,k
so that we still have a trivalence graph. This requires that the
support dimension Dii⬘ⴱ of the fixed-point wave function
Using Eq. 共27兲 and its variation
⎟
⎠.
and
j
⎜
Φfix ⎝
j
k i
α
Nkiiⴱ N jⴱ jkⴱ
via the first type of renormalization steps 共see Fig. 10兲,
which are local unitary transformations. In the simplified
second type renormalization, we want to reduce
i
共25兲
We will call such a wave function renormalization step a
␣␤
satisfies
P-move.64 Here Pkj,
i
k i’
α
β
⎟
⎠
⎞
i
i
i
The simplified second type renormalization can now be written as 共since Diiⴱ = 1兲
⎛
⎞
β j ⎟
⎜
kj,αβ
i
Φfix ⎝
Φfix
.
⎠ Pi
can be reduced to
β
j
k i’
α
Dii⬘ⴱ = ␦ii⬘ .
But we can define a simpler second type renormalization,
by noting that
i
α
c
b
γ
β
a
β
is given by
⎞
i
j
⎞
i
m
⎟
⎟ P lm,µν
Φfix
i∗
⎠
we can rewrite Eq. 共31兲 as
155138-12
i
.
共32兲
PHYSICAL REVIEW B 82, 155138 共2010兲
LOCAL UNITARY TRANSFORMATION, LONG-RANGE…
␣␤
ei␪P1 Pkj,
=
i
ⴱ
ⴱ
jj k,␤␣
i mj,␭␥ lm,␮␯
,
Fi i m ,␭␥Fm i l,␯␮ Pi
兺
m,␭,␥,l,␯,␮
ⴱⴱ
ⴱⴱ ⴱ
ⴱ
␣␤
.
which is a condition on Pkj,
i
ijm,␣␤
␣␤
can be obtained by
More conditions on Fkln,␹␦ and Pkj,
i
noticing that
⎛
⎞
⎛
⎞
p
p
η
⎜i j
Φfix ⎜
⎝ αm β
k
N ⎟ ⎜i
ijm,αβ
⎟
⎜
F
Φ
fix
kln,χδ
⎠
⎝
n=0 χ,δ
l
which implies that
⎛
Pijp,αη δim Φfix ⎝
i
β
k
η k
⎟
j χ ⎟,
⎠
n
δ
l
共34兲
共38兲
The condition 兺iAi共Ai兲ⴱ = 1 is simply the normalization conⴱ
dition of the wave function. The condition Ai = Ai come from
the fact that the graph
i
can be deformed into the graph
i
⎞
⎠
l
⎛
ijm,αβ jp,χη
Fkln,χδ
Pk∗ δkn Φfix ⎝
n,χ,δ
i
δ
l
k
⎞
⎠.
共35兲
We find
on a sphere.
To find the conditions that determine Ai, let us first consider the fixed-point wave function where the index on a link
is i:
Γ
i
,
Φfix (i, Γ) = Φfix
where ⌫ are indices on other part of graph. We note that the
graph
Nkjkⴱ
ei␪P2 Pijp,␣␩␦im␦␤␦ =
兺i Ai共Ai兲ⴱ = 1.
ⴱ
Ai = Ai ,
共33兲
ijm,␣␤ jp,␹␩
兺 Fklk,
␹␦ P k
ⴱ
␹=1
Γ
for all k,i,l satisfying Nkilⴱ ⬎ 0.
共36兲
i
can be deformed into the graph
i
Γ
E. Fixed-point wave functions from the fixed-point gLU
transformations
on a sphere. Thus
In the last section, we discussed the conditions that a
ijm,␣␤
kj,␣␤
兲 must satisfy.
fixed-point gLU transformation 共Fkln,
␥␭ , P i
After finding a fixed-point gLU transformation
ijm,␣␤
kj,␣␤
兲, in this section, we are going to discuss how
共Fkln,
␥␭ , P i
to calculate the corresponding fixed-point wave function ⌽fix
from the solved fixed-point gLU transformation
ijm,␣␤
kj,␣␤
兲.
共Fkln,
␥␭ , P i
First we note that, using the two types of wave function
renormalization introduced above, we can reduce any graph
to
i
So, once we know
,
i
Here A satisfy
i
= Ai = Ai
∗
i
= Φfix
i
to i
Γ
Φfix
Γ
i
i
= Φfix
Γ
i
Γ
.
:
f (i, Γ)Φfix
i
共39兲
Φfix
we can reconstruct the full fixed-point wave function ⌽fix on
any connected graph.
Let us assume that
Φfix
Γ
Using the F-moves and the P-moves, we can reduce
i
We see that
.
Φfix
Φfix
i
= Ai = 0
共40兲
for all i. Otherwise, any wave function with i link will be
zero.
To find more conditions on Ai, we note that
⎛
⎞
j
λ
⎜
⎟
mj,γλ
m ⎟
Φfix ⎜
Φfix
= Pimj,γλ Ai .
i
⎝ γ ⎠ Pi
i
共37兲
By rotating
155138-13
共41兲
PHYSICAL REVIEW B 82, 155138 共2010兲
CHEN, GU, AND WEN
i
i , j , k , ␣ that satisfy Nkji ⬎ 0 and for any nonzero vector v␣,
i
j
α
β
α vα Φfix
j
λ
m
γ
k
ⴱⴱ
␥␭ i
by 180°, we can show that Pmj,
A ⯝ Pmjⴱ i
i
,␭␥ j
A or
mⴱiⴱ,␭␥ j
␥␭ i
Pmj,
A = ei␪A1 P jⴱ
i
共42兲
A.
We also note that
⎛
j
Φfix ⎝ α
⎛
⎞
k
β⎠
i
m,λ,γ
⎜
∗
,αβ
⎜
Fjijk
∗ im,λγ Φfix
⎝
i
λ
m
γ
j
⎞
is nonzero for some ␤. This means that the matrix M kji is
invertible, where M kji is a matrix whose elements are given
by
i
⎟
⎟.
⎠
Let us define
␪
i␪⬘
⌽ikj,
␣␤ = e
␥␭ i
F j im,␭␥ Pmj,
A,
兺
i
m,␭,␥
ⴱ
where
⎛
.
k
共44兲
Φfix
i
α j
β
= det(Mkji ),
k
⎞
j
k
Φfix ⎝ α
Φθikj,αβ , ≡
β
共43兲
This allows us to show
ijkⴱ,␣␤
α j
(Mkji )αβ = Φfix
β⎠
we find that
i
Φθikj,αβ = e i θA2 Φθkji,αβ ,
Φθikj,αβ = 0, if Nikj = 0.
⎛
det Φfix
⎝α j
β⎠
k
共45兲
␪
␪
The condition ⌽ikj,
␣␤ ⯝ ⌽kji,␣␤ comes from the fact that the
graph
⎞
i
= det[Φθkji,αβ ] = 0.
共46兲
The above also implies that
j
α k
β
i
共47兲
Nkji = Niⴱ jⴱkⴱ .
and the graph
i
α j
β
k
can be deformed into each other on a sphere.
Also, for any given i , j , k , ␣ that satisfy Nkji ⬎ 0, the wave
function
i
Φfix
α j
must be nonzero for some ⌫, where ⌫ represents indices on
other part of the graph. Then after some F-moves and
P-moves, we can reduce
α j
Φfix
to
α j
k
= Φfix
α j
β
k
ijm,␣␤
kj,␣␤
leads to some equations for Fkln,
, and Ai. More
␥␭ , P i
ijm,␣␤
kj,␣␤
i
equations for Fkln,␥␭ , Pi , and A , can be obtained by using
the relations
⎛
⎞
⎛
⎞
n
χ
l
⎜
Φfix ⎝
β.
α j
δ
l
i j
αm
β
k
So, for any given i , j , k , ␣ that satisfy Nkji ⬎ 0,
i
Φfix
β
i
i
Γ
α k
Γ
k
i
The conditions 关Eqs. 共38兲, 共42兲, 共44兲, and 共45兲兴 allow us to
␪
determine Ai 共and ⌽ikj,
␣␤兲.
From Eq. 共45兲, we see that relation
j
i
k
⎟
⎜
⎠ = Φfix ⎝
⎛
⎜
= Φfix ⎝
β
k
is nonzero for some ␤. Since such a statement is true for any
choices of basis on the vertex ␣, we find that for any given
β
k
α
m i
αj
χ
n
j
i n
m δl k
χ
from the tetrahedron rotation symmetry63,65 and
155138-14
δ
β
⎟
⎠
⎞
⎟
⎠,
共48兲
PHYSICAL REVIEW B 82, 155138 共2010兲
LOCAL UNITARY TRANSFORMATION, LONG-RANGE…
⎛
⎜
Φfix ⎝
δ
l
n
i j
αm
χ
k
⎞
⎛
⎟
⎠
β
⎜
ijm ,αβ
Fkln,γλ
Φfix ⎝ l
λ
γλ
n
δ
∗
i
χ
k⎟
⎠
j
n
⎞
γ
⎛
⎜
ijm∗ ,αβ ln∗ i∗ ,δλ
Fkln,γλ
Fnlp∗ ,σ Φfix ⎝ l
γλ,pσ
ε
p
σ
χ
γ
n
∗
∗ ∗
ijm ,αβ ln i ,δλ pl
Fkln,γλ
Fnlp∗ ,σ Pn
∗
,σ
∗
∗ ∗
ijm ,αβ ln i ,δλ pl
Fkln,γλ
Fnlp∗ ,σ Pn
∗
,σ
Φfix ⎝
⎞
k⎟
⎠
j
⎛
γλ,pσ
=
n
χ
j
n
⎞
k⎠
γ
Φθkjn∗ ,γχ .
γλ,pσ
It is not clear if Eqs. 共48兲 and 共49兲 will lead to new independent equations or not. In the following discussions, we will
not include Eqs. 共48兲 and 共49兲. We find that, at least for
simple cases, the equations without Eqs. 共48兲 and 共49兲 are
enough to completely determine the solutions.
To summarize, the conditions 关Eqs. 共9兲, 共12兲, 共18兲, 共23兲,
共30兲, 共33兲, 共36兲, 共38兲, 共40兲, 共42兲, and 共44兲–共47兲兴 form a set of
ijm,␣␤
kj,␣␤
,
nonlinear equations whose variables are Nijk, Fkln,
␥␭ , P i
␣␤
␣␤
ijm,
kj,
Ai, and 共␪F , ␪ P1 , ␪ P2兲. Finding Nijk, Fkln,␥␭ , Pi , and Ai that
satisfy such a set of nonlinear equations corresponds to finding a fixed-point gLU transformation that has a nontrivial
fixed-point
wave
function.
So
the
solutions
ijm,␣␤
kj,␣␤
i
,
P
,
A
兲
give
us
a
characterization
of topo共Nijk , Fkln,
␥␭
i
logical orders. This may lead to a classification of topological order from the local unitary transformation point of view.
IX. SIMPLE SOLUTIONS OF THE FIXED-POINT
CONDITIONS
In this section, let us find some simple
fixed-point conditions 关Eqs. 共9兲, 共12兲, 共18兲,
共36兲, 共38兲, 共40兲, 共42兲, and 共44兲–共47兲兴 for the
ijm,␣␤
kj,␣␤
兲 and
transformations 共Nijk , Fkln,
␥␭ , P i
i
wave function A .
solutions of the
共23兲, 共30兲, 共33兲,
fixed-point gLU
the fixed-point
A. Unimportant phase factors in the solutions
Formally, the solutions of the fixed-point conditions are
not isolated. They are parametrized by several continuous
phase factors. In this section, we will discuss the origin of
those phase factors. We will see that those different phase
factors do not correspond to different states of matter 共i.e.,
different equivalence classes of gLU transformations兲. So after removing those unimportant phase factors, the solutions
of the fixed-point conditions are isolated 共at least for the
simple examples studied here兲.
We notice that, apart from two normalization conditions,
␣␤
and Ai.
all of the fixed-point conditions are linear in Pkj,
i
ijm,␣␤
kj,␣␤
i
, A 兲 is a solution, then
Thus if 共Fkln,␥␭ , Pi
ijm,␣␤ i␾1 kj,␣␤ i␾2 i
, e A 兲 is also a solution. However, the
共Fkln,
␥␭ , e P i
two phase factors ei␾1,2 do not lead to different fixed-point
共49兲
wave functions since they only affect the total phase of the
wave function and are unphysical. Thus the total phases of
i
Pkj
i and A can be adjusted. We can use this degree of freeⱖ 0 and A0 ⬎ 0.
dom to set, say, P00,11
0
ijm,␣␤
Similarly the total phase of Fkln,
␥␭ is also unphysical and
can be adjusted. We have used this degree of freedom to
reduce Eq. 共17兲 to Eq. 共18兲. But this does not totally fix the
ijm,␣␤
ijm,␣␤
ijm,␣␤
total phase of Fkln,
␥␭ . The transformation Fkln,␥␭ → −Fkln,␥␭
does not affect Eq. 共18兲. We can use such a transformation to
ijm,␣␤
set the real part of a nonzero component of Fkln,
␥␭ to be
positive.
The above three phase factors are unphysical. However,
the fixed-point solutions may also contain phase factors that
do correspond to different fixed-point wave functions. For
ˆ
example, the local unitary transformation ei␪l0M l0 does not
affect the fusion rule Nijk, where M̂ l0 is the number of links
with 兩l0典 state and 兩lⴱ0典 state. Such a local unitary transformaijm,␣␤
kj,␣␤
, Ai兲 and generates a continuous
tion changes 共Fkln,
␥␭ , P i
family of the fixed-point wave functions parametrized by ␪l0.
Those wave functions are related by local unitary transformations that continuously connect to identity. Thus, those
fixed-point wave functions all belong to the same phase.
Similarly, we can consider the following local unitary
共i0 j0k0兲
兩␤典 that acts on each vertex
transformation 兩␣典 → 兺␤U␣␤
with states 兩i0典 , 兩j0典 , 兩k0典 on the three edges connecting to the
vertex. Such a local unitary transformation also does not affect the fusion rule Nijk. The new local unitary transformaijm,␣␤
kj,␣␤
, Ai兲 and generates a continuous
tion changes 共Fkln,
␥␭ , P i
family of the fixed-point wave functions parametrized by the
共i0 j0k0兲
unitary matrix U␣␤
. Again, those fixed-point wave functions all belong to the same phase.
In the following, we will study some simple solutions of
the fixed-point conditions. We find that, for those examples,
the solutions have no addition continuous parameter apart
from the phase factors discussed above. This suggests that
the solutions of the fixed-point conditions correspond to isolated zero-temperature phases.
B. N = 1 loop state
Let us first consider a system where there are only two
states 兩0典 and 兩1典 on each link of the graph. We choose iⴱ
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= i and the simplest fusion rule that satisfies Eqs. 共9兲, 共30兲,
and 共47兲 is
p 1A 0 = p 2A 1,
␩ p 3A 1 = p 1A 0,
兩A0兩2 + 兩A1兩2 = 1, 共53兲
where ␩ = ⫾ 1. The above simplified equations can be solved
exactly. We find two solutions parametrized by ␩ = ⫾ 1,
N000 = N110 = N101 = N011 = 1,
other
共50兲
Nijk = 0.
f 0 = f 1 = f 2 = 1,
Since Nijk ⱕ 1, there is no states on the vertices. So the indices ␣ , ␤ , . . . labeling the states on a vertex can be suppressed.
The above fusion rule corresponds to the fusion rule for
the N = 1 loop state discussed in Ref. 63. So we will call the
corresponding graphic state N = 1 loop state.
Due to the relation 关Eq. 共18兲兴, the different components of
ijm
are not independent. There are only four inthe tensor Fkln
dependent potentially nonzero components which are denoted as f 0 , . . . , f 3:
000
F000
= f0
000
F111
011
= (F100
101
)∗ = (F010
110
= F001
= f1
011
F011
101
= (F101
) ∗ = f2
110
F110
= f3
P01
0 = p 1,
)∗
共51兲
P00
1 = p 2,
P01
1 = p3 .
共52兲
We can adjust the total phases of pi and Ai to make p0
ⱖ 0 and A0 ⱖ 0. We can also use the local unitary transforˆ
mation ei␪l0M l0 with l0 = 1 to make f 1 ⱖ 0 since the F’s described by f 1 in Eq. 共51兲 are the only F’s that change the
number of 兩1典 links.
The fixed-point conditions 关Eqs. 共9兲, 共12兲, 共18兲, 共23兲, 共30兲,
共33兲, 共36兲, 共38兲, 共40兲, 共42兲, and 共44兲–共47兲兴 form a set of nonlinear equations on the ten variables f i, pi, and Ai. Many of
the nonlinear equations are dependent or even equivalent.
Using a computer algebraic system, we simplify the set of
nonlinear equations. The simplified equations are 共after making the phase choice described above兲
f 0 = f 1 = f 2 = 1,
f3 = ␩,
p 1 = p 3 = ␩ p 0,
p2 = p0 ,
p20 + 兩p1兩2 = 1,
冑2 ,
A0 =
冑2 ,
1
p1 = p3 =
A1 =
␩
冑2 ,
␩
冑2 .
共54兲
We also find
ijm
in Eq. 共11兲 relates wave functions on two
We note that Fkln
graphs. In the above we have drawn the two related graphs
after the F tensor, where the first graph following F corresponds to the graph on the left-hand side of Eq. 共11兲 and the
second graph corresponds to the graph on the right-hand side
of Eq. 共11兲. The doted line corresponds to the 兩0典 state on the
link and the solid line corresponds to the 兩1典 state on the link.
There are four potentially nonzero components in Pkj
i , which
are denoted by p0 , . . . , p3,
P00
0 = p 0,
1
p0 = p2 =
f3 = ␩,
兩p2兩2 + 兩p3兩2 = 1,
ei␪F = ei␪P1 = ei␪P2 = ei␪A1 = ei␪A2 = 1.
共55兲
The ␩ = 1 fixed-point state corresponds to the Z2 loop condensed state whose low-energy effective-field theory is the
Z2 gauge theory.63,66 We call such a state, simply, the Z2
state. The ␩ = −1 fixed-point state corresponds to the doublesemion state whose low-energy effective-field theory is the
U共1兲 ⫻ U共1兲 Chern-Simons gauge theory63,66
L=
1
共2a1␮⳵␯a1␭⑀␮␯␭ − 2a2␮⳵␯a2␭⑀␮␯␭兲.
4␲
共56兲
C. N = 1 string-net state
To obtain another class of simple solutions, we modify the
fusion rule to
N000 = N110 = N101 = N011 = N111 = 1,
other
Nijk = 0
共57兲
while keeping everything the same. The above Nijk also satisfies Eqs. 共9兲, 共30兲, and 共47兲.
The new fusion rule corresponds to the fusion rule for the
N = 1 string-net state discussed in Ref. 63. So we will call the
corresponding graphic state N = 1 string-net state.
Again, due to the relation 关Eq. 共18兲兴, the different compoijm
are not independent. Now there are
nents of the tensor Fkln
seven independent potentially nonzero components which
are denoted as f 0 , . . . , f 6,
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000
F000
= f0
000
F111
011
= (F100
101
)∗ = (F010
110
= F001
= f1
011
F011
101
= (F101
) ∗ = f2
011
F111
101
= (F111
111
)∗ = F011
111
= (F101
) ∗ = f3
110
F110
= f4
110
F111
111
= (F110
F 111
= f6
)∗
The fixed-point state corresponds to the N = 1 string-net
condensed state63 whose low-energy effective-field theory is
the doubled SO共3兲 Chern-Simons gauge theory.66
D. An N = 2 string-net state—The Z3 state
The above simple examples correspond to nonorientable
string-net states. Here we will give an example of orientable
string-net state. We choose N = 2, 0ⴱ = 0, 1ⴱ = 2, 2ⴱ = 1, and
N000 = N012 = N120 = N201 = N021 = N102 = N210 = N111 = N222
) ∗ = f5
共58兲
The above Nijk satisfies Eqs. 共9兲, 共30兲, and 共47兲.
Due to the relation 关Eq. 共18兲兴, the different components of
ijm
are not independent. There are eight indepenthe tensor Fkln
dent potentially nonzero components which are denoted as
f 0 , . . . , f 7:
Note that F’s described by f 1 and f 5 are the only F’s that
change the number of 兩1典 links and the number of 兩1典兩1典兩1典
vertices. So we can use the local unitary transformation
ˆ
ˆ
ei共␪M 1+␾M 111兲 to make f 1 and f 5 to be positive real numbers.
共Here M̂ 1 is the total number of 兩1典 links and M̂ 111 is the total
number of 兩1典兩1典兩1典 vertices.兲 We also use the freedom of
ijm
to make Re共f 0兲 ⱖ 0.
adjusting the total sign of Fkln
There are five potentially nonzero components in Pkj
i ,
which are denoted by p0 , . . . , p4,
P00
0 = p 0,
P01
0 = p 1,
P01
1 = p 3,
p21 f 24 + p21 = 1,
p 0 = f 4 p 1,
P00
1 = p2 ,
A 0 = f 4A 1,
p 3 = p 1,
p0 = p2 =
␥
,
␥2 + 1
f 4 = − f 6 = ␥,
A0 =
p1 = p3 =
␥
,
␥ +1
2
A1 =
011
= (F200
120
)∗ = F002
202
= (F020
) ∗ = f1
022
= (F100
101
)∗ = (F010
210
= F001
= f2
022
= F022
101
= (F202
202
= (F101
) ∗ = f3
101
= (F121
112
)∗ = F021
112
= (F102
) ∗ = f4
202
= (F212
221
)∗ = F012
221
= (F201
) ∗ = f5
120
= (F221
210
)∗ = (F112
221
= F120
= f6
210
= (F220
) ∗ = f7
011
F122
022
F211
112
F210
p4 = 0,
f 24 + f 4 − 1 = 0.
Let ␥ be the positive solution of ␥2 + ␥ = 1: ␥ =
that f 5 = 冑␥. The above can be written as
f 0 = f 1 = f 2 = f 3 = 1,
000
F111
共59兲
f 4 = f 25 = − f 6 ⬎ 0,
共A0兲2 + 共A1兲2 = 1,
= f0
011
F011
P11
1 = p4 .
p 2 = p 0,
000
F000
000
F222
We use the freedom of adjusting the total phase of Pkj
i to
make p0 to be a positive number. We can also use the freedom of adjusting the total phase of Ai to make A0 to be a
positive number.
The fixed-point conditions 关Eqs. 共12兲, 共18兲, 共23兲, 共33兲,
共36兲, 共38兲, 共40兲, 共42兲, and 共44兲–共46兲兴 form a set of nonlinear
equation on the variables f i, pi, and Ai, which can be simplified. The simplified equations have the following form:
f 0 = f 1 = f 2 = f 3 = 1,
共63兲
= 1.
1
,
␥2 + 1
冑5−1
2
120
F110
共60兲
. We see
1
.
␥ +1
2
P00
0 = p 0,
We also find
ei␪F = ei␪P1 = ei␪P2 = ei␪A1 = ei␪A2 = 1.
P01
0 = p 1,
P02
1 = p 5,
共61兲
共62兲
)∗
)∗
共64兲
There are nine potentially nonzero components in
are denoted by p0 , . . . , p8,
f 5 = 冑␥ ,
p4 = 0,
)∗
P02
0 = p 2,
P00
2 = p 6,
P00
1 = p 3,
P01
2 = p 7,
Pkj
i ,
which
P01
1 = p4 ,
P02
2 = p8 .
共65兲
Using the transformations discussed in Sec. IX B, we can fix
the phases of f 1, f 2, f 6, and p0 to make them positive.
The fixed-point conditions 关Eqs. 共12兲, 共18兲, 共23兲, 共33兲,
共36兲, 共38兲, 共40兲, 共42兲, and 共44兲–共46兲兴 form a set of nonlinear
equation on the variables f i, pi, and Ai, which can be solved
exactly. After fixing the phases using the transformations discussed in Sec. IX B, we find only one solution
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f i = 1,
pi =
1
冑3 ,
i = 0,1, . . . ,7,
i = 0,1, . . . ,8,
A0 = A1 = A2 =
1
冑3 .
共66兲
We also find
ei␪F = ei␪P1 = ei␪P2 = ei␪A1 = ei␪A2 = 1.
共67兲
The fixed-point state corresponds to the Z3 string-net condensed state63 whose low-energy effective-field theory is the
U共1兲 ⫻ U共1兲 Chern-Simons gauge theory55,63
L=
1
共3a1␮⳵␯a2␭⑀␮␯␭ + 3a2␮⳵␯a1␭⑀␮␯␭兲,
4␲
共68兲
which is the Z3 gauge theory.
We note that all the above simple solutions also satisfy the
standard pentagon identity, although we solved the weaker
projective pentagon identity. It is not clear if we can find
nontrivial solutions that do not satisfy the standard pentagon
identity.
X. A CLASSIFICATION OF TIME-REVERSAL INVARIANT
TOPOLOGICAL ORDERS
There are several ways to define time-reversal operation
for the graphic states. The simplest one is given by
T̂:⌽共⌫兲 → ⌽ⴱ共⌫兲,
共69兲
where ⌫ represents the labels on the vertices and links which
are not changed under T̂. 共This corresponds to the situation
where the different states on the links and the vertices are
realized by different occupations of scalar bosons.兲 For such
a time reversal transformation, T̂2 = 1 and the real solutions
of the fixed-point conditions 关Eqs. 共9兲, 共12兲, 共18兲, 共23兲, 共30兲,
共33兲, 共36兲, 共38兲, 共40兲, 共42兲, and 共44兲–共46兲兴 give us a classification of time reversal invariant topological orders in local
spin systems. Note that the time-reversal invariant topological orders are equivalent class of local orthogonal transformations that connect to the identity transformation continuously.
ijm,␣␤
kj,␣␤
, Ai兲 of the
Different real solutions 共Nijk , Fkln,
␥␭ , P i
fixed-point conditions do not always correspond to different
time-reversal invariant topological orders. The solutions differ by some unimportant phase factors 共which are ⫾1 signs兲
correspond to the same topological order.
To understand the above result, we notice that, from the
ijm,␣␤
kj,␣␤
, A i兲
structure of the fixed-point conditions, if 共Fkln,
␥␭ , P i
ijm,␣␤
kj,␣␤
i
, ␩AA 兲 is also a soluis a solution, then 共␩FFkln,␥␭ , ␩ P Pi
tion, where ␩F = ⫾ 1, ␩ P = ⫾ 1, and ␩A = ⫾ 1. However, the
three phase factors ␩F, ␩ P, and ␩A do not lead to different
fixed-point wave functions since they only affect the total
phase of the wave function and are unphysical.
On the other hand, the fixed-point solutions may also contain phase factors that do correspond to different fixed-point
wave functions. For example, the local orthogonal transforˆ
mation ei␲M l0 does not affect the fusion rule Nijk, where M̂ l0
is the number of links with 兩l0典 state and 兩lⴱ0典 state. Such a
ijm,␣␤
kj,␣␤
, A i兲
local orthogonal transformation changes 共Fkln,
␥␭ , P i
and generates a discrete family of the fixed-point wave functions.
Similarly, we can consider the following local orthogonal
共i0 j0k0兲
transformation 兩␣典 → 兺␤O␣␤
兩␤典 that acts on each vertex
with states 兩i0典 , 兩j0典 , 兩k0典 on the three edges connecting to the
vertex. Such a local orthogonal transformation also does not
affect the fusion rule Nijk. The new local orthogonal transforijm,␣␤
kj,␣␤
, Ai兲 and generates a family of
mation changes 共Fkln,
␥␭ , P i
the fixed-point wave functions parametrized by the orthogo共i0 j0k0兲
nal matrix O␣␤
.
Now the question is that do those solutions related by
local orthogonal transformations have the same time-reversal
invariant topological order or not. We know that two gapped
wave functions have the same time-reversal invariant topological order if and only if they can be connected by local
orthogonal transformation that connects to identity continuously. It is well known that an orthogonal matrix whose determinant is −1 does not connect to identity. Thus it appears
that local orthogonal transformations some times can generate different time reversal invariant topological orders.
However, when we use the equivalent classes of local
orthogonal transformations to define time-reversal invariant
topological orders, we not only assume the local orthogonal
transformations to connect to identity continuously, we also
assume that we can expand the local Hilbert spaces 共say by
increasing the range of the indices i and ␣ that label the
states on the edges and the vertices兲. The local orthogonal
transformations can act on those enlarged Hilbert spaces and
can connect to identity in those enlarged Hilbert spaces.
Even when a local orthogonal transformation cannot be deformed into identity in the original Hilbert space, it can always be deformed into identity continuously in an enlarged
Hilbert space. Thus two real wave functions related by a
local orthogonal transformation always have the same timereversal invariant topological order.
1 0
兲 that acts on
For example, an orthogonal matrix 共 0−1
states 兩0典 and 兩1典 does not connect to identity within the
space of two by two orthogonal matrices. However, we can
embed the above orthogonal matrix into a three by three
orthogonal matrix that acts on 兩0典, 兩1典, and 兩2典:
冢
1
0
0 −1
0
0
0
0
−1
冣
.
Such a three by three orthogonal matrix does connect to
identity within the space of three by three orthogonal matrices. This completes our argument that all local orthogonal
transformations can connect to identity continuously at least
in an enlarged Hilbert space.
So, after factoring out the unimportant phase factors discussed above, the real solutions of the fixed-point conditions
may uniquely correspond to time-reversal invariant topological orders. The four types of real solutions discussed in the
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last section are examples of four different time-reversal invariant topological orders.
For the N = 1 loop states and the N = 1 string-net state, we
only have two states on each link. In this case, we can treat
the two states as the two states of an electron spin. The
time-reversal transformation now becomes
T̂:c0兩0典 + c1兩1典 → − cⴱ1兩0典 + cⴱ0兩1典
共70兲
on each link. For such a time-reversal transformation T̂2 =
−1. The two N = 1 loop states and the N = 1 string-net state
are not invariant under such a time-reversal transformation.
XI. WAVE FUNCTION RENORMALIZATION
FOR TENSOR PRODUCT STATES
A. Motivation
Once the fixed-point states have been identified and the
labeling of topological orders has being found, we then face
the next important issue: given a generic ground-state wave
function of a system, how to identify the topological orders
in the state? In other words, how to calculate the data
ijm,␣␤
kj,␣␤
, Ai兲 that characterize the topological or共Nijk , Fkln,
␥␭ , P i
ders from a generic wave function?
One way to address the above issue is to have a general
renormalization procedure which flows other states in the
same phase to the simple fixed-point state so that we can
identify topological order from the resulting fixed-point
state. That is, we want to find a local unitary transformation
which removes local entanglement and gets rid of unnecessary degrees of freedom from the state. How to find the
appropriate unitary transformation for a specific state is then
the central problem in this renormalization procedure. Such a
procedure for one-dimensional tensor product states 共TPS兲
共also called matrix product states兲 has been given in Ref. 42.
Here we will propose a method to renormalize twodimensional TPS, where nontrivial topological orders
emerge. The basic idea is to use the gLU transformation
discussed in Sec. VII. Note that, through the gLU transformation, we can reduce the number of labels in a region A to
the minimal value without loosing any quantum information
共see Fig. 6兲. This is because the gLU transformation is a
lossless projection into the support space of the state in the
region A. By performing such gLU transformations on overlapping regions repeatedly 关see Fig. 2共a兲兴, we can reduce a
generic wave function to the simple fixed-point form discussed above. It should be noted that any state reducible in
this way can be represented as a MERA.43
In the following, we will present this renormalization procedure for two-dimensional TPS where we find a method to
calculate the proper gLU transformations. The tensor product
states are many-body entangled quantum states described
with local tensors. By making use of the entanglement information contained in the local tensors, we are able to come up
with an efficient algorithm to renormalize two-dimensional
TPS.
This algorithm can be very useful in the study of quantum
phases. Due to the efficiency in representation, TPS has
found wide application as variational ansatz states in the
FIG. 11. 共Color online兲 Left: tensor T representing a 2D quantum state on hexagonal lattice. i is the physical index and ␣ , ␤ , ␥ are
the inner indices. Right: a tensor product state where each vertex is
associated with a tensor. The inner indices of the neighboring tensors connect according to the underlying hexagonal lattice.
studies of quantum many-body systems.67–73 Suppose that in
a variational study we have found a set of tensors which
describe the ground state of a two-dimensional many-body
Hamiltonian and want to determine the phase this state belongs to. We can apply our renormalization algorithm to this
tensor product state, which removes local entanglement and
flows the state to its fixed point. By identifying the kind of
order present in the fixed-point state, we can obtain the phase
information for the original state.
In this section, we will give a detailed description of the
algorithm and in the next section we will present its application to some simple 共but nontrivial兲 cases. The states we are
concerned with have translational symmetry and can be described with a translational invariant tensor network. To be
specific, we discuss states on a hexagonal lattice. Generalization to other regular lattices is straightforward.
B. Tensor product states
Consider a two-dimensional spin model on a hexagonal
lattice with one spin 共or one qudit兲 living at each vertex. The
Hilbert space of each spin is D dimensional. The state can be
represented by assigning to every vertex a set of tensors
i
, where i 共see Fig. 11兲 labels the local physical dimenT␣␤␥
sion and takes value from 1 to D. ␣ , ␤ , ␥ are inner indices
along the three directions in the hexagonal lattice, respectively. The dimension of the inner indices is d. Note that the
figures in this note are all sideviews with inner indices in the
horizontal plane and the physical indices pointing in the vertical direction, if not specified otherwise.
The wave function is given in terms of these tensors by
兩␺典 =
兺
tTr共Ti1Ti2, . . . ,Tim, . . .兲兩i1i2, . . . ,im, . . .典,
i1,i2,. . .,im,. . .
共71兲
where tTr denotes tensor contraction of all the connected
inner indices on the links of the hexagonal lattice.
A renormalization procedure of quantum states is composed of local unitary transformations and isometry maps
such that the state flows along the path 兩␺共0兲典 , 兩␺共1兲典 , 兩␺共2兲典 , . . .
and finally toward a fixed point 兩␺⬁典. With the tensor product
representation, flow of states corresponds to a flow of tensors
T共0兲 , T共1兲 , T共2兲 , . . . We will give the detailed procedure of how
the tensors are mapped from one step to the next in the
following section.
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FIG. 12. 共Color online兲 Double tensor T represented as two
layers of tensor T with the physical indices contracted. The gray
layer is the lower layer.
C. Renormalization algorithm
FIG. 13. 共Color online兲 F-move in the renormalization procedure: 共1兲 combining double tensors T1 and T2 on neighboring sites
into a single double tensor T, 共2兲 splitting double tensor T into
tensor T̃, and 共3兲 SVD decomposition of tensor T̃ into tensors Ta
and Tb.
In one round of renormalization, we start from tensor T共n兲,
do some operation to it which corresponds to local unitary
transformations on the state, and map T共n兲 to T共n+1兲. The
whole procedure can be broken into two parts: the F-move
and the P move, in accordance with the two steps introduced
in the previous section.
Note that with respect to the bipartition of indices
␣⬘␤⬘␦⬘⑀⬘ and ␣␤␦⑀, T is Hermitian
1. Step 1: F-move
T␣⬘␤⬘␦⬘⑀⬘,␣␤␦⑀ = 共T␣␤␦⑀,␣⬘␤⬘␦⬘⑀⬘兲ⴱ
T␣⬘␤⬘␦⬘⑀⬘,␣␤␦⑀ =
兺 T1,␣⬘␤⬘␥⬘,␣␤␥ ⫻ T2,␦⬘⑀⬘␥⬘,␦⑀␥ .
共73兲
␥⬘,␥
共74兲
and positive semidefinite. Therefore it has a spectral decomposition with positive eigenvalues 兵␭ j ⱖ 0其. The corresponding eigenvectors are 兵T̂ j其,
In the F-move, we take a
j
T␣⬘␤⬘␦⬘⑀⬘,␣␤␦⑀ = 兺 ␭ j共T̂␣⬘␤⬘␦⬘⑀⬘兲ⴱ ⫻ T̂␣␤
␦⑀ .
j
configuration in the tensor network and map it to a
configuration by doing a local unitary operation. We will see
that the tensor product representation of a state leads to a
natural way of choosing an appropriate unitary operation for
the renormalization of the state.
In order to do so, first we define the double tensor T of
tensor T as
i
.
T␣⬘␤⬘␥⬘,␣␤␥ = 兺 共T␣⬘␤⬘␥⬘兲ⴱ ⫻ T␣␤␥
i
共75兲
j
共72兲
i
Graphically the double tensor T is represented by two layers
of tensor T with the physical indices connected 共see Fig. 12兲.
The tensor T giving rise to the same double tensor T is not
unique. Any tensor T⬘ which differs from T by an unitary
transformation U on physical index i gives the same T as U
and U† cancels out in the contraction of i. On the other hand,
an unitary transformation on i is the only degree of freedom
possible, i.e., any T⬘ which gives rise to the same T as T
differs from T by a unitary on i. Therefore, in the process of
turning a tensor T into a double tensor T and then split it
again into a different tensor T⬘, we apply a nontrivial local
unitary operation on the corresponding state. A well designed
way of splitting the double tensor will give us the appropriate unitary transformation we need, as we show below.
F-move has the following steps. First, construct double
tensors for two neighboring sites on the lattice and combine
them into a single double tensor with eight inner indices.
This spectral decomposition lead to a special way of decomposing double tensor T into tensors. Define a rank 8 tensor
T̃共as shown in Fig. 13 after step 2兲 as follows:
lmnr
冑 j ⴱ j
T̃␣␤
␦⑀ = 兺 ␭ j共T̂lmnr兲 ⫻ T̂␣␤␦⑀ .
共76兲
j
T̃ has four inner indices ␣ , ␤ , ␦ , ⑀ of dimension d and four
physical indices l , m , n , r also of dimension d which are in
the direction of ␣ , ␤ , ␦ , ⑀, respectively. As 兵T̂ j其 form an orthonormal set, it is easy to check that T̃ gives rise to double
tensor T. Going from T1 and T2 to T̃, we have implemented
a local unitary transformation on the physical degrees of
freedom on the two sites so that in T̃ the physical indices and
the inner indices represent the same configuration. In some
sense, we are keeping only the physical degrees of freedom
necessary for entanglement with the rest of the system while
getting rid of those that are only entangled within this local
region. Now we do a singular value decomposition of tensor
T̃ in the direction orthogonal to the link between T1 and T2
and T̃ is decomposed into tensors Ta and Tb,
ln
mr
lmnr
T̃␣␤
␦⑀ = 兺 Ta,␣␦␭ ⫻ Tb,␤⑀␭ .
␭
共77兲
This step completes the F-move. Ideally, this step should be
done exactly so we are only applying local unitary operations
to the state. Numerically, we keep some large but finite cut-
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FIG. 16. 共Color online兲 The corner double line tensor which is a
fixed point of the renormalization algorithm. The three groups of
indices 兵␣ , ␤ , i1其, 兵␥ , ␦ , i2其, and 兵⑀ , ␭ , i3其 are entangled within each
group but not between the groups.
FIG. 14. Original hexagonal lattice 共gray line兲 and renormalized
lattice 共black line兲 after F-move has been applied to the neighboring
pairs of sites circled by dash line.
off dimension for the singular value decomposition step, so
this step is approximate.
On a hexagonal lattice, we do F-move on the chosen
neighboring pairs of sites 共dash circled in Fig. 14兲 so that the
tensor network is changed into a configuration shown by
thick dark lines in Fig. 14. Physical indices are omitted from
this figure. Now by grouping together the three tensors that
meet at a triangle, we can map the tensor network back into
a hexagonal lattice, with 1/3 the number of sites in the original lattice. This is achieved by the P-move introduced in the
next section.
2. Step 2: P-move
Now we contract the three tensors that meet at a triangle
together to form a new tensor in the renormalized lattice as
shown in the first step in Fig. 15,
pq
I
nr
lm
T␣␤␥
= 兺 Ta,
␣␦⑀ ⫻ Tb,␤␭␦ ⫻ Tc,␥⑀␭ ,
␦⑀␭
共78兲
where I is the physical index of the new tensor which includes all the physical indices of Ta , Tb , Tc: l , m , n , r , p , q.
Note that in the contraction, only inner indices are contracted
and the physical indices are simply group together. Constructing the double tensor T from T, we get the renormalized double tensor on the new hexagonal lattice which is in
FIG. 15. 共Color online兲 P-move in the renormalization procedure: 共1兲 contracting three tensors that meet at a triangle Ta , Tb , Tc
to form a new tensor T共1兲 on one site of the renormalized hexagonal
lattice. 共2兲 Constructing the double tensor T共1兲 from T共1兲 so that we
can start to do F-moves again.
the same form as T1 , T2 and we can go back again and do the
F-move.
3. Complications: Corner double line
One problem with the above renormalization algorithm is
that, instead of having one isolated fixed-point tensor for
each phase, the algorithm has a continuous family of fixed
points which all correspond to the same phase. Consider a
tensor with structure shown in Fig. 16. The tensor is a tensor
product of three parts which include indices 兵␣ , ␤ , i1其,
兵␥ , ␦ , i2其, and 兵⑀ , ␭ , i3其, respectively. It can be shown that this
structure remains invariant under the renormalization flow.
Therefore, any tensor of this structure is a fixed point of our
renormalization flow. However, it is easy to see that the state
it represents is a tensor product of loops around each
plaquette, which can be disentangled locally into a trivial
product state. Therefore, the states all have only short-range
entanglement and correspond to the topologically trivial
phase. The trivial phase has then a continuous family of
fixed-point tensors. This situation is very similar to that discussed in Refs. 41 and 74. We will keep the terminology and
call such a tensor a corner double line tensor. Not only does
corner double line tensor complicate the situation in the
trivial phase, it leads to a continuous family of fixed points in
every phase. It can be checked that the tensor product of a
corner double line with any other fixed-point tensor is still a
fixed-point tensor. The states they correspond to differ only
by small loops around each plaquette and represent the same
topological order. Therefore any single fixed-point tensor
gets complicated into a continuous class of fixed-point tensors. In practical application of the renormalization algorithm, in order to identify the topological order of the fixedpoint tensor, we need to get rid of such corner double line
structures. Due to their simple structure, this can always be
done, as discussed in the next section.
XII. APPLICATIONS OF THE RENORMALIZATION
FOR TENSOR PRODUCT STATES
Now we present some examples where our algorithm is
used to determine the phase of a tensor product state. The
algorithm can be applied both to symmetry breaking phases
and topological ordered phases. In the study of symmetry
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CHEN, GU, AND WEN
breaking/topological ordered phases, suppose that we have
obtained some tensor product description of the ground state
of the system Hamiltonian. We can then apply our algorithm
to the tensors, flow them to the fixed point, and see whether
they represent a state in the symmetry-breaking phase/
topological ordered phase or a trivial phase.
For system with symmetry/topological order related to
gauge symmetry, it is very important to keep the symmetry/
gauge symmetry in the variational approach to ground state
and search within the set of tensors that have this symmetry/
gauge symmetry.75 The resulting tensor will be invariant under such symmetries/gauge symmetries but the state they
correspond to may have different orders. In the symmetrybreaking case, the state could have this symmetry or could
spontaneous break it. In the topological ordered phase, the
state could have nontrivial topological order or be just trivial.
Our algorithm can then be applied to decide which is the
case. In order to correctly determine the phase for such symmetric tensors, it is crucial that we maintain the symmetry/
gauge symmetry of the tensor throughout our renormalization process. We will discuss in detail two cases: the Ising
symmetry-breaking phase and the Z2 topological ordered
phase. For simplicity of discussion and to demonstrate the
generality of our renormalization scheme, we will first introduce the square lattice version of the algorithm. All subsequent applications are carried out on square lattice. 共Algorithm on a hexagonal lattice would give qualitatively similar
result, though quantitatively they might differ, e.g., on the
position of critical point.兲
FIG. 17. 共Color online兲 Renormalization procedure on square
lattice part 1: 共1兲 decomposing double tensor T into tensor T̃ and 共2兲
SVD decomposition of T̃ in two different directions, resulting in
tensors Ta , Tb and Tc , Td, respectively.
tion of tensor T̃. For vertices in sublattice A we decompose
between the up-right and down-left directions as shown in
step 2 of Fig. 17. For vertices in sublattice B we decompose
between the up-left and down-right directions as shown in
step 3 of Fig. 17,
lr
mn
lmnr
T̃␣␤␥
␦ = 兺 Ta,␣␦␭ ⫻ Tb,␤␥␭ ,
␭
lr
mn
lmnr
T̃␣␤␥
␦ = 兺 Tc,␣␦␭ ⫻ Tb,␤␥␭ .
␭
A. Renormalization on square lattice
Tensor product states on a square lattice are represented
i
with one tensor T␣␤␥
␦ on each vertex, where i is the physical
index and ␣␤␥␦ are the four inner indices in the up, down,
left, right directions, respectively. We will assume translational invariance and require the tensor to be the same on
every vertex. The renormalization procedure is be a local
unitary transformation on the state which flows the form of
the tensor until it reaches the fixed point. It is implemented
in the following steps. First, we form the double tensor T
from tensor T,
After the decomposition, the original lattice 共gray lines in
Fig. 18兲 is transformed into the configuration shown by thick
dark lines in Fig. 18. Physical indices are omitted from this
figure. If we now shrink the small squares, we get a tensor
product state on a renormalized square lattice. Figure 19
shows how this is done.
In step 1, we contract the four tensors that meet at a small
square
i
T␣⬘␤⬘␥⬘␦⬘,␣␤␥␦ = 兺 共T␣⬘␤⬘␥⬘␦⬘兲ⴱ ⫻ T␣␤␥
␦.
i
i
Then do the spectral decomposition of positive operator T
into
j
T␣⬘␤⬘␥⬘␦⬘,␣␤␥␦ = 兺 ␭ j共T̂␣⬘␤⬘␥⬘␦⬘兲ⴱ ⫻ T̂␣␤␥
␦
j
j
lmnr
and form a new tensor T̃␣␤␥
␦
lmnr
冑 j ⴱ j
T̃␣␤␥
␦ = 兺 ␭ j共T̂lmnr兲 ⫻ T̂␣␤␥␦ ,
j
where l , m , n , r are physical indices in the up, down, left, and
right directions, respectively. This is illustrated in step 1 of
Fig. 17. This step is very similar to the second step in the
F-move on hexagonal lattice. Next we do SVD decomposi-
FIG. 18. Original square lattice 共gray line兲 and renormalized
lattice 共black line兲 after the operations in Fig. 17 has been applied.
Physical indices of the tensors are not drawn here.
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a simple form of tensor and demonstrate how the algorithm
works.
Suppose that the tensors obtained from the variational
i
study T␣␤␥
␦, where i , ␣ , ␤ , ␥ , ␦ can be 0 or 1, takes the following form:
0
␣+␤+␥+␦
T␣␤␥
,
␦=␭
1
4−共␣+␤+␥+␦兲
T␣␤␥
.
␦=␭
共79兲
␭ is a parameter between 0 and 1. Under an X operation to
the physical index, the tensor is changed to T̃,
0
4−共␣+␤+␥+␦兲
,
T̃␣␤␥
␦=␭
FIG. 19. 共Color online兲 Renormalization procedure on square
lattice part 2: 共1兲 combining four tensors that meet at a square into
a single tensor and 共2兲 constructing a double tensor from it.
I
T␣␤␥
␦=
m n
m n
m n
m n
Ta,
兺
␤␯␹ ⫻ Tb,␣␭␮ ⫻ Tc,␦␮␹ ⫻ Td,␥␭␯ ,
␭␮␯␹
1 1
3 3
2 2
4 4
where I stands for the combination of all physical indices
mi , ni, i = 1 , . . . , 4. In step 2, we construct a double tensor T
form tensor T and completes one round of renormalization.
Now we can go back to step 1 in Fig. 17 and flow the tensor
further.
Now we are ready to discuss two particular examples, the
Ising symmetry-breaking phase and the Z2 topological ordered phase, to demonstrate how our algorithm can be used
to determine the phase of a tensor product state.
B. Ising symmetry-breaking phase
A typical example for symmetry-breaking phase transition
is the transverse field Ising model. Consider a square lattice
with one spin 1/2 on each site. The transverse field Ising
model is
HIsing = 兺 ZiZ j + ⑀ 兺 Xk ,
ij
k
where 兵ij其 are nearest-neighbor sites. The Hamiltonian is invariant under spin-flip transformation 兿kXk for any ⑀.
When ⑀ = 0, the ground state spontaneously breaks this
symmetry into either the all spin-up state 兩00¯ 0典 or the all
spin-down state 兩11¯ 1典. In this case any global superposition ␣兩00¯ 0典 + ␤兩11¯ 1典 represents a degenerate ground
state. When ⑀ = ⬁, the ground state has all spin polarized in
the X direction 共兩++ ¯ +典兲 and does not break this symmetry.
In the variational study of this system, we can require that
the variational ground state always have this symmetry, regardless if the system is in the symmetry-breaking phase or
not. Then we will find for ⑀ = 0 the ground state to be
兩00¯ 0典 + 兩11¯ 1典. Such a global superposition represents
the spontaneous symmetry breaking. For ⑀ = ⬁, we will find
the ground state to be 兩++ ¯ +典 and does not break the symmetry. For 0 ⬍ ⑀ ⬍ ⬁, we will need to decide which of the
previous two cases it belongs to. We can first find a tensor
product representation of an approximate ground state which
is symmetric under the spin-flip transformation, then apply
the renormalization algorithm to find the fixed point and decide which phase the state belongs to. Below we will assume
1
␣+␤+␥+␦
.
T̃␣␤␥
␦=␭
T̃ can be mapped back to T by switching the 0,1 label for the
four inner indices ␣␤␥␦. Such a change in basis for the inner
indices does not change the contraction result of the tensor
and hence the state that is represented. Therefore, the state is
invariant under the spin-flip transformation 兿kXk and we will
say that the tensor has this symmetry also.
When ␭ = 0, the tensor represents state 兩00¯ 0典
+ 兩11¯ 1典, which corresponds to the spontaneous symmetrybreaking phase. We note that the ␭ = 0 tensor is a direct sum
of dimension-1 tensors. Such a direct-sum structure corresponds to spontaneous symmetry breaking, as discussed in
detail in Ref. 41. When ␭ = 1, the tensor represents state 兩+
+ ¯ +典 which corresponds to the symmetric phase. When 0
⬍ ␭ ⬍ 1, there must be a phase transition between the two
phases. However, as ␭ goes from 0 to 1, the tensor varies
smoothly with well-defined symmetry. It is hard to identify
the phase-transition point. Now we can apply our algorithm
to the tensor. First, we notice that at ␭ = 0 or 1, the tensor is
a fixed point for our algorithm. Next, we find that for ␭
⬍ 0.358, the tensor flows to the form with ␭ = 0 while for ␭
⬎ 0.359, it flows to the form with ␭ = 1. Therefore, we can
clearly identify the phase a state belongs to using this algorithm and find the phase-transition point.
Note that in our algorithm, we explicitly keep the spin-flip
symmetry in the tensor. That is, after each renormalization
step, we make sure that the renormalized tensor is invariant
under spin-flip operations up to change in basis for the inner
indices. If the symmetry is not carefully preserved, we will
not be able to tell the two phases apart.
We also need to mention that for arbitrary ␭, the fixed
point that the tensor flows to can be different from the tensor
at ␭ = 0 or 1 by a corner double line structure. We need to get
rid of the corner double line structure in the result to identify
the real fixed point. This is possible by carefully examining
the fixed-point structure. Another way to distinguish the different fixed points without worrying about corner double
lines is to calculate some quantities from the fixed-point tensors that are invariant with the addition of corner double
lines. We also want the quantity to be invariant under some
trivial changes to the fixed point, such as a change in scale
T → ␩T or the change in basis for physical and inner indices.
One such quantity is given by the ratio of X2 and X1
defined
as
X1 = 共兺␣⬘␥⬘,␣␥T␣⬘␣⬘␥⬘␥⬘,␣␣␥␥兲2
and
X2
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T
T
T
T
FIG. 20. Quantity X2 / X1 obtained by taking the ratio of the
contraction value of the double tensor in two different ways. X2 / X1
is invariant under change in scale, basis transformation and corner
double line structures of the double tensor and can be used to distinguish different fixed-point tensors. For clarity, only one layer of
the double tensor is shown. The other layer connects in exactly the
same way.
= 兺␣⬘␤⬘␥⬘␦⬘,␣␤␥␦T␣⬘␣⬘␥⬘␦⬘,␣␣␥␦ ⫻ T␤⬘␤⬘␦⬘␥⬘,␤␤␦␥. Figure 20
gives a graphical representation of these two quantities. In
this figure, only one layer of the double tensor is shown. The
other layer connects in the exactly the same way. It is easy to
verify that X2 / X1 is invariant under the change in scale, basis
transformation, and corner double lines.
We calculate X2 / X1 along the renormalization flow. The
result is shown in Fig. 21. At the ␭ = 0 fixed point,
X2 / X1 = 0.5 while at ␭ = 1, X2 / X1 = 1. As we increase the
number of renormalization steps, the transition between the
two fixed points becomes sharper and finally approaches a
step function with critical point at ␭c = 0.358. Tensors with
␭ ⬍ ␭c belongs to the symmetry-breaking phase while tensors
with ␭ ⬎ ␭c belongs to the symmetric phase.
C. Z2 topological ordered phase
The algorithm can also be used to study topological order
of quantum states. In this section, we will demonstrate how
the algorithm works with Z2 topological order.
Consider again a square lattice but now with one spin 1/2
per each link. A simple Hamiltonian on this lattice with Z2
topological order can be defined as
H Z2 = 兺
兿 Xi + 兺 j苸兿 Z j ,
p i苸p
v
v
where p means plaquettes and i 苸 p is all the spin 1/2’s
around the plaquette and v means vertices and j 苸 v is all the
FIG. 21. 共Color online兲 X2 / X1 for tensors 关Eq. 共79兲兴 under the
renormalization flow. As the number of RG steps increases, the
transition in X2 / X1 becomes sharper and finally approaches a step
function at fixed point. The critical point is at ␭c = 0.358.
spin 1/2’s connected to the vertex. The ground-state wave
function of this Hamiltonian is a fixed-point wave function
and corresponds to the N = 1 loop state with ␩ = 1 as discussed in the previous section.
The ground-state wave function has a simple tensor product representation. For simplicity of discussion we split every spin 1/2 into two and associate every vertex with four
ijkl
spins. The tensor T␣␤␥
␦,Z2 has four physical indices i , j , k , l
= 0 , 1 and three inner indices ␣ , ␤ , ␥ , ␦ = 0 , 1.
ijkl
Tijkl,Z
= 1,
2
=0
if mod共i + j + k + l,2兲
all other terms being 0.
It can be checked that TZ2 is a fixed-point tensor of our
algorithm. This tensor has a Z2 gauge symmetry. If we apply
Z operation to all the inner indices, where Z maps 0 to 0 and
1 to −1, the tensor remains invariant as only even configurations of the inner indices are nonzero in the tensor.
Consider then the following set of tensor parametrized by
g
i+j+k+l
Tijkl
,
ijkl = g
=0
if mod共i + j + k + l,2兲
all other terms being 0.
共80兲
At g = 1, this is exactly TZ2 and the corresponding state has
topological order. At g = 0, the tensor represents a product
state of all 0 and we denote the tensor as T0. At some critical
point in g, the state must go through a phase transition. This
set of tensors are all invariant under gauge transformation
ZZZZ on their inner indices and the tensor seems to vary
smoothly with g. One way to detect the phase transition is to
apply our algorithm. We find that, at g ⬎ gc, the tensors flow
to TZ2 while at g ⬍ gc, the tensors flow to T0. We determine gc
to be between 0.804 and 0.805. As this model is mathematically equivalent to two-dimensional classical Ising model
where the transition point is known to great accuracy, we
compare our result to that result and find our result to be
within 1% accuracy 共gc = 0.8022兲. Again in the renormalization algorithm, we need to carefully preserve the Z2 gauge
symmetry of the tensor so that we can correctly determine
the phase of the states.
The fixed-point tensor structure might also be complicated by corner double line structures but it is always possible to identify and get rid of them. Similarly, we can calculate the invariance quantity X2 / X1 to distinguish the two
fixed points. X2 / X1 = 1 for TZ2 while X2 / X1 = 0.5 for T0. The
result is plotted in Fig. 22 and we can see that the transition
in X2 / X1 approaches a step function after a large number of
steps of RG, i.e., at the fixed point. The critical point is at
gc = 0.804. For g ⬍ gc, the tensor belongs to the trivial phase
while for g ⬎ gc, the tensor belongs to the Z2 topological
ordered phase.
Our algorithm can also be used to demonstrate the stability of topological order against local perturbation. As is
shown in Ref. 75, local perturbations to the Z2 Hamiltonian
correspond to variations in tensor that do not break the Z2
gauge symmetry. We picked tensors in the neighborhood of
TZ2 which preserve this gauge symmetry randomly and applied our renormalization algorithm 共gauge symmetry is kept
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jkn,␹␦
ijm,␣␤ ⴱ
共Fkln,
␹␦ 兲 = Flⴱiⴱmⴱ,␤␣ ,
ijm⬘,␣⬘␤⬘ ijm,␣␤ ⴱ
共Fkln,␹␦ 兲 = ␦m␣␤,m⬘␣⬘␤⬘ ,
兺 Fkln,
␹␦
n,␹,␦
Nkjtⴱ Ntinⴱ Nltsⴱ
兺t ␩兺=1 ␸兺=1 ␬兺=1
Nqmpⴱ
ijm,␣␤ itn,␸␹ jkt,␩␬
Fknt,
␩␸ Flps,␬␥Flsq,␦␾
=e
i␪F
兺
⑀=1
mkn,␤␹ ijm,␣⑀
Flpq,
␦⑀ Fqps,␾␥ ,
共82兲
␣␤
=
ei␪P1 Pkj,
i
ⴱ
ⴱ
␥ lm,␮␯
,
兺 Fijji k,m␤␣,␭␥Fmi mj,␭
i l,␯␮ Pi
m,␭,␥,l,␯,␮
ⴱⴱ ⴱ
ⴱⴱ
ⴱ
Nkjkⴱ
FIG. 22. 共Color online兲 X2 / X1 for tensors 关Eq. 共80兲兴 under the
renormalization flow. As the number of RG steps increases, the
transition in X2 / X1 becomes sharper and finally approaches a step
function at fixed point. The critical point is at gc = 0.804.
throughout the renormalization process兲. We find that as long
as the variation is small enough, the tensor flows back to TZ2,
up to a corner double line structure. This result shows that
the Z2 topological ordered phase is stable against local perturbations.
e
i␪ P2
Pijp,␣␩␦im␦␤␦
=
ijm,␣␤ jp,␹␩
兺 Fklk,
␹␦ P k
for all k,i,l satisfying Nkilⴱ ⬎ 0
共83兲
关see Eqs. 共9兲, 共12兲, 共18兲, 共23兲, 共30兲, 共33兲, 共36兲, and 共47兲兴.
ijm,␣␤
kj,␣␤
兲 we can further find out
From the data 共Nijk , Fkln,
␹␦ , P i
the fixed-point wave function by solving the following equations for Ai , i = 0 , . . . , N:
ⴱ
Ai = ei␪AAi ⫽ 0,
XIII. SUMMARY
In this paper, we discuss a defining relation between local
unitary transformation and quantum phases. We argue that
two gapped states are related by a local unitary transformation if and only if the two states belong to the same quantum
phase.
We can use the equivalent classes of local unitary transformations to define “patterns of long range entanglement.”
So the patterns of long-range entanglement correspond to
universality classes of quantum phases, and are the essence
of topological orders.32
As an application of this point of view of quantum phases
and topological order, we use the generalized local unitary
transformations to generate a wave function renormalization,
where the wave functions flow within the same universality
class of a quantum phase 共or the same equivalent class of the
local unitary transformations兲. In other words, the renormalization flow of a wave function does not change its topological order. Such a wave function renormalization allows us to
classify topological orders, by classifying the fixed-point
wave functions and the associated fixed-point local unitary
transformations.
First, we find that the fixed-point local unitary transforijm,␣␤
kj,␣␤
兲 that
mations are described by the data 共Nijk , Fkln,
␹␦ , P i
satisfy
N
兺 Nii kN jk j
Nijk = N jki = Nkⴱ jⴱiⴱ ⱖ 0,
ⴱ
ⴱ ⴱ
ⱖ 1,
j,k=0
N
N
兺 N jim Nkml = n=0
兺 Nkjn Nl ni ,
ⴱ
m=0
ⴱ
ⴱ
ⴱ
共81兲
ⴱ
␹=1
兺i Ai共Ai兲ⴱ = 1,
mⴱiⴱ,␭␥ j
␥␭ i
A = ei␪A1 P jⴱ
Pmj,
i
␪
i␪⬘
⌽ikj,
␣␤ = e
A,
ⴱ
␥␭ i
A,
兺 Fijkj im,␭,␣␤␥Pmj,
i
m,␭,␥
ⴱ
␪
i␪A1 ␪
⌽kji,␣␤ ,
⌽ikj,
␣␤ = e
␪
⌽ikj,
␣␤ = 0,
if Nikj = 0,
␪
det共⌽ikj,
␣␤兲 ⫽ 0
共84兲
关see Eqs. 共38兲, 共40兲, 共42兲, and 共44兲–共46兲兴.
ijm,␣␤
kj,␣␤
, Ai兲 that satisfy the
The combined data 共Nijk , Fkln,
␹␦ , P i
conditions 关Eqs. 共81兲–共84兲兴 classify a large class of topological orders. We see that the problem of classifying a large
class of topological orders becomes the problem of solving a
set of nonlinear algebraic equations 关Eqs. 共81兲–共84兲兴.
ijm,␣␤
kj,␣␤
, Ai兲 that satisfy the
The combined data 共Nijk , Fkln,
␹␦ , P i
conditions 关Eqs. 共81兲–共84兲兴 also classify a large class of
time-reversal invariant topological orders, if we restrict ourselves to real solutions. The solutions related by local orthogonal transformations all belong to the same phase since
the local orthogonal transformations always connect to identity if we enlarge the Hilbert space.
We like to point out that we cannot claim that the soluijm,␣␤
kj,␣␤
, Ai兲 classify all topological orders
tions 共Nijk , Fkln,
␹␦ , P i
since we have assumed that the fixed-point local unitary
transformations are described by tensors of finite dimensions. It appears that chiral topological orders, such as quantum Hall states, are described by tensors of infinite dimensions.
155138-25
PHYSICAL REVIEW B 82, 155138 共2010兲
CHEN, GU, AND WEN
ijm,␣␤
kj,␣␤
We note that the data 共Nijk , Fkln,
, Ai兲 just charac␹␦ , P i
terize different fixed-point wave functions. It is not guaranteed that the different data will represent different topological orders. However, for the simple solutions discussed in
this paper, they all coincide with string-net states, where the
topological properties, such as the ground-state degeneracy,
number of quasiparticle types, the quasiparticle statistics,
etc., were calculated from the data. From those topological
properties, we know that those different simple solutions repijm,␣␤
kj,␣␤
, A i兲
resent different topological orders. Also 共Fkln,
␹␦ , P i
are real for the simple solutions discussed here. Thus they
also represent topological orders with time-reversal symmetry. 共Certainly, at the same time, they represent stable topological orders even without time-reversal symmetry.兲
We also like to point out that our description of fixedpoint local unitary transformations is very similar to the description of string-net states. However, the conditions 关Eqs.
共81兲–共84兲兴 on the data appear to be weaker than 共or equivalent to兲 those63 on the string-net data. So the fixed-point
wave functions discussed in this paper may include all the
string-net states 共in 2D兲.
Last, we present a wave function renormalization scheme,
based on the gLU transformations for generic TPS. Such a
wave function renormalization always flows within the same
phase 共or within the same equivalence class of LU transformations兲. It allows us to determine which phase a generic
TPS belongs to by studying the resulting fixed-point wave
functions. We demonstrated the effectiveness of our method
for both symmetry-breaking phases and topological ordered
phases. We find that we can even use tensors that do not
break symmetry to describe spontaneous symmetry-breaking
states: if a state described by a symmetric tensor has a spontaneous symmetry breaking, the symmetric tensor will flow
to a fixed-point tensor that has a form of direct sum.
ACKNOWLEDGMENTS
We would like to thank I. Chuang, M. Hastings, M. Levin,
F. Verstraete, Z.-H. Wang, Y.-S. Wu, and S. Bravyi for some
very helpful discussions. X.G.W. is supported by NSF under
Grant No. DMR-0706078. Z.C.G. is supported in part by the
NSF under Grant No. NSFPHY05-51164.
APPENDIX: EQUIVALENCE RELATION BETWEEN
QUANTUM STATES IN THE SAME PHASE
In Secs. III and IV, we argued about the equivalence relation between gapped quantum ground states in the same
phase. We concluded that two states are in the same phase if
and only if they can be connected by local unitary evolution
or constant depth quantum circuit,
兩⌽共1兲典 ⬃ 兩⌽共0兲典
iff 兩⌽共1兲典 = T关e−i兰0dgH共g兲兴兩⌽共0兲典,
兩⌽共1兲典 ⬃ 兩⌽共0兲典
1
˜
M
iff 兩⌽共1兲典 = Ucirc
兩⌽共0兲典.
Now we want to make these arguments more rigorous, by
stating clearly what is proved and what is conjectured, and
by giving precise definition of two states being the same, the
locality of operators, etc. We will show the equivalence in
the following steps: 共1兲 if two gapped ground states are in the
same phase, then they are connected by local unitary evolution, 共2兲 a gapped ground state remains in the same phase
under local unitary evolution, and 共3兲 a local unitary evolution can be simulated by a constant depth quantum circuit
and vice versa. 共All these discussions can be generalized to
the case where the system has certain symmetries. H̃共g兲 and
Ucirc used in the equivalence relation will then have the same
symmetry as the system Hamiltonian H.兲
First, according to the definition in Sec. III, two states
兩⌽共0兲典 and 兩⌽共1兲典 are in the same phase if we can find a
family of local Hamiltonians H共g兲, g 苸 关0 , 1兴 with 兩⌽共g兲典
being its ground state such that the ground-state average of
any local operator O, 具⌽共g兲兩O兩⌽共g兲典 changes smoothly from
g = 0 to g = 1. Here we allow a more general notion of locality
for the Hamiltonian34 and require H共g兲 to be a sum of local
operators HZ共g兲,
H共g兲 =
兺 HZ共g兲,
共A1兲
Z苸Z
where HZ共g兲 is a Hermitian operator defined on a compact
region Z. 兺Z苸Z sums over a set Z of regions. The set Z
contains regions that differ by translations. The set Z also
contains regions with different sizes. However, HZ共g兲 approaches zero exponentially as the size of the region Z approaches infinity. Or more precisely, for all sites u in the
lattice
兺
储HZ共g兲储兩Z兩exp关␮diam共Z兲兴 = O共1兲
共A2兲
Z苸Z,Z苹u
for some positive constant ␮. Here 兺Z苸Z,Z苹u sums over all
regions in the set Z that cover the site u, 储 ¯ 储 denotes operator norm, 兩Z兩 is the cardinality of Z, and diam共Z兲 is the
diameter of Z. Therefore, instead of being exactly zero outside of a finite region, the interaction terms can have an
exponentially decaying tail.
If 兩⌽共0兲典 and 兩⌽共1兲典 are gapped ground states of H共0兲 and
H共1兲, then we assume that for all 具⌽共g兲兩O兩⌽共g兲典 to be
smooth, H共g兲 must remain gapped for all g. If H共g兲 closes
gap for some gc, then there must exist a local operator O
such that 具⌽共g兲兩O兩⌽共g兲典 has a singularity at gc. We call the
gapped H共g兲 an adiabatic connection between two states in
the same phase.
The existence of an adiabatic connection gives rise to a
local unitary evolution between 兩⌽共0兲典 and 兩⌽共1兲典. A slight
modification of Lemma 7.1 in Ref. 34 gives theorem 1 which
is given below.
Theorem 1. Let H共g兲 be a differentiable family of local
Hamiltonians and 兩⌽共g兲典 be its ground state. If the excitation
gap above 兩⌽共g兲典 is larger than some finite value ⌬ for all g
, then we can define H̃共g兲 = i兰dtF共t兲exp关iH共g兲t兴关⳵gH共g兲兴exp关
1 ˜
−iH共g兲t兴 , such that 兩⌽共1兲典 = T关e−i兰0dgH共g兲兴兩⌽共0兲典, where F共t兲
is a function which has the following properties: 共1兲 the fourier transform of F共t兲, F̃共␻兲 is equal to −1 / ␻ for 兩␻兩 ⬎ ⌬. 共2兲
F̃共␻兲 is infinitely differentiable. 共3兲 F共t兲 = −F共−t兲. Under this
definition, H̃共g兲 is local 共almost兲 and satisfies
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PHYSICAL REVIEW B 82, 155138 共2010兲
LOCAL UNITARY TRANSFORMATION, LONG-RANGE…
where h共l兲 decays faster than any negative power of l. Therefore, HZ共g兲 remains local up to a tail which decays faster
than any negative power. H共g兲 then forms a local gapped
adiabatic connection between 兩⌽共0兲典 and 兩⌽共1兲典.
To detect for phase transition, we must check whether the
ground-state average value of any local operator O,
具⌽共g兲兩O兩⌽共g兲典, has a singularity or not. 具⌽共g兲兩O兩⌽共g兲典
= 具⌽共0兲兩U†gOUg兩⌽共0兲典. Using the Lieb-Robinson bound
given in Ref. 34, we find U†gOUg remains local 关in the sense
of Eq. 共87兲兴 and evolves smoothly with g. Therefore,
具⌽共g兲兩O兩⌽共g兲典 changes smoothly. In fact, because H共g兲 is
differentiable, the derivative of 具⌽共g兲兩O兩⌽共g兲典 to any order
always exists. Therefore, there is no singularity in the
ground-state average value of any local operator O and
兩⌽共0兲典 and 兩⌽共1兲典 are in the same phase. We have hence
shown that states connected with local unitary evolution are
in the same phase.
This completes the equivalence relation stated in 共1兲. The
definition of locality is slightly different in different cases, so
there is still some gap in the equivalence relation. However,
we believe that the equivalence relation should be valid with
a slight generalization of the definition of locality.
Lastly, we want to show that the equivalence relation is
still valid if we use constant depth quantum circuit instead of
local unitary evolution. This is true because we can always
simulate a local unitary evolution using a constant depth
quantum circuit and vice versa.
To simulate a local unitary evolution of the form
1 ˜
T关e−i兰0dgH共g兲兴, first divide the total time into small segments
共m+1兲␦t ˜
␦t such that T关e−i兰m␦t dgH共g兲兴 ⯝ e−i␦tH共m␦t兲. H共m␦t兲
= 兺ZHZ共m␦t兲. The set of local operators 兵HZ共m␦t兲其 can always be divided into a finite number of subsets 兵HZi 共m␦t兲其,
1
兵HZi 共m␦t兲其¯ such that elements in the same subset com2
mute with each other. Then we can do Trotter expansion and
approximate e−i␦tH共m␦t兲 as e−i␦tH共m␦t兲 ⯝ 共U共1兲U共2兲¯兲n, where
U共1兲 = 兿ie−i␦tHZi1/n, U共2兲 = 兿ie−i␦tHZi2/n¯ Each term in U共1兲, U共2兲
commute, therefore they can be implement as a piecewise
local unitary operator. Putting these piecewise local unitary
operators together, we have a quantum circuit which simulates the local unitary evolution. The depth of circuit is proportional to n ⫻ 共1 / ␦t兲. It can be shown that to achieve a
simulation with constant error, a constant depth circuit would
suffice.76
On the other hand, to simulate a constant depth quantum
共2兲
共2兲
共k兲
共k兲
circuit Ucirc = U共1兲
pwlU pwl ¯ U pwl, where U pwl = 兿iUi with a local unitary evolution, we can define the time-dependent
Hamiltonian as H共t兲 = 兺iH共k兲
i , such that for 共k − 1兲␦t ⬍ t ⬍ k␦t,
共k兲
M ␦t ˜
.
It
is
easy
to
check that T关e−i兰0 dgH共g兲兴 = Ucirc.
eiHi ␦t = U共k兲
i
The simulation time needed is M ␦t. We can always choose a
finite ␦t such that 储H共k兲
i 储 is finite and the evolution time is
finite.
We have shown that constant depth quantum circuit and
local unitary evolution can simulate each other. The equivalence relation 关Eq. 共1兲兴 can therefore also be stated in terms
of constant depth circuit as in Eq. 共4兲.
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储关H̃Z共g兲,OB兴储 ⱕ h⬘关dist共Z,B兲兴兩Z兩储H̃Z共g兲储储OB储,
共A3兲
where OB is any operator supported on site B, dist共Z , B兲 is
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is a function which decays faster than any negative power of
l. 关 ¯ 兴 denotes commutator of two operators. This is called
the quasiadiabatic continuation of states.
Therefore, we can show that if 兩⌽共0兲典 and 兩⌽共1兲典
are in the same phase, then we can find a local 关as defined
in Eq. 共87兲兴 Hamiltonian H̃Z共g兲 such that 兩⌽共1兲典
1 ˜
= T关e−i兰0dgH共g兲兴兩⌽共0兲典. In other words, states in the same
phase are equivalent under local unitary evolution.
Note that here we map 兩⌽共0兲典 exactly to 兩⌽共1兲典. There is
another version of quasiadiabatic continuation,35 where the
mapping is approximate. In that case, H̃Z共g兲 can be defined
to have only exponentially small tail outside of a finite region instead of a tail which decays faster than any negative
1 ˜
power. T关e−i兰0dgH共g兲兴兩⌽共0兲典 will not be exactly the same as
兩⌽共1兲典 but any local measurement on them will give approximately the same result.
Next we want to show that the reverse is also true. Suppose that 兩⌽共0兲典 is the gapped ground state of a local Hamiltonian H共0兲, H共0兲 = 兺ZHZ共0兲 and each HZ共0兲 is supported on
a finite region Z. Apply a local unitary evolution generated
by H̃共s兲 to 兩⌽共0兲典 and take it to 兩⌽共g兲典, g = 0 ⬃ 1, 兩⌽共g兲典
g ˜
= Ug兩⌽共0兲典, where Ug = T关e−i兰0dsH共s兲兴. 兩⌽共g兲典 is then ground
state of H共g兲 = UgH共0兲U†g = 兺ZUgHZ共0兲U†g = 兺ZHZ共g兲. Under
unitary transformation Ug the spectrum of H共0兲 does not
change, therefore H共g兲 remains gapped. To show that H共g兲
also remains local, we use the Lieb-Robinson bound derived
in Ref. 34, which gives
储关HZ共g兲,OB兴储 = 储关UgHZ共0兲U†g,OB兴储
ⱕ h关dist共Z,B兲兴兩Z兩储HZ共0兲储储OB储,
1 L.
2 V.
6 I.
7 D.
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PHYSICAL REVIEW B 82, 155138 共2010兲
CHEN, GU, AND WEN
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